Quadratic Equations: Introduction and Solving Methods

Introduction and Terms Related to Quadratic Equations

Beyond Linear Equations

Having explored linear equations, which are polynomial equations of the first degree, we now advance to the next level of polynomial equations: those of the second degree. These are known as quadratic equations. Quadratic equations and the functions associated with them are fundamental in algebra and have widespread applications in numerous fields such as physics (e.g., projectile motion), engineering (e.g., bridge design), economics (e.g., optimizing profit), and even everyday life (e.g., calculating areas).

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Definition of a Quadratic Equation

A quadratic equation in one variable is a polynomial equation in which the highest power of the variable is exactly $2$. The general form, known as the standard form, of a quadratic equation in the variable $x$ is:

In this standard form:

• $x$: This is the single variable for which we are trying to find a value(s).
• $a, b,$ and $c$: These are real coefficients, meaning they are constants that can be any real number (integers, fractions, decimals, irrational numbers).
• $a$: This is the coefficient of the $x^2$ term. It is also called the leading coefficient. For the equation to be quadratic, the coefficient $a$ must be non-zero ($a \neq 0$). If $a$ were $0$, the term $ax^2$ would disappear, and the equation would reduce to $bx+c=0$, which is a linear equation (degree 1), not a quadratic one (degree 2).
• $b$: This is the coefficient of the $x$ term. It can be any real number, including zero ($b=0$).
• $c$: This is the constant term. It does not have the variable $x$. It can also be any real number, including zero ($c=0$).

Examples of quadratic equations in standard form or easily convertible to standard form:

• $x^2 - 5x + 6 = 0$: Here $a=1, b=-5, c=6$.
• $2y^2 + 4y = 0$: Here $a=2, b=4, c=0$. (Constant term is 0).
• $3m^2 - 7 = 0$: Here $a=3, b=0, c=-7$. (Linear term coefficient is 0).
• $z^2 = 9$: Rearrange to $z^2 - 9 = 0$. Here $a=1, b=0, c=-9$.
• $(x+1)^2 = 4$: Expand LHS: $x^2 + 2x + 1 = 4$. Rearrange to standard form: $x^2 + 2x + 1 - 4 = 0 \implies x^2 + 2x - 3 = 0$. Here $a=1, b=2, c=-3$.

Examples of equations that are NOT quadratic equations in one variable:

• $x^3 - 2x + 1 = 0$: The highest power of $x$ is $3$. This is a cubic equation.
• $x + 5 = 0$: The highest power of $x$ is $1$. This is a linear equation.
• $x^2 + y^2 = 10$: This equation involves two different variables, $x$ and $y$.
• $\sqrt{x} + x^2 = 0$: The term $\sqrt{x}$ means $x^{1/2}$, which is not a polynomial term (exponent is not a non-negative integer).
• $\frac{1}{x^2} + 1 = 0$: The term $\frac{1}{x^2}$ means $x^{-2}$, which has a negative exponent. This is not a polynomial equation.

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Roots of a Quadratic Equation

The roots (or solutions) of a quadratic equation $ax^2 + bx + c = 0$ are the specific numerical value(s) of the variable $x$ that make the equation a true statement when substituted into it. These values are also called the zeros of the corresponding quadratic polynomial $P(x) = ax^2 + bx + c$. Finding the roots is the primary objective when solving a quadratic equation.

According to the Fundamental Theorem of Algebra, a polynomial equation of degree $n \ge 1$ with complex coefficients has exactly $n$ complex roots, when counted with multiplicity. For a quadratic equation (degree $n=2$), this means it always has exactly two roots in the complex number system. These two roots can take different forms depending on the values of the coefficients $a, b,$ and $c$:

• Two distinct real roots: The equation has two different real numbers as solutions. This happens when the discriminant ($b^2 - 4ac$) is positive.
• One real root with multiplicity 2 (two equal real roots): The equation has exactly one real number as a solution, but it is counted twice. This happens when the discriminant ($b^2 - 4ac$) is zero. The graph of the corresponding polynomial touches the x-axis at this root.
• Two complex conjugate roots: The equation has two roots that are complex numbers (involving the imaginary unit $i$), and these roots are conjugates of each other (e.g., $p+qi$ and $p-qi$). These roots are not real numbers. This happens when the discriminant ($b^2 - 4ac$) is negative. The graph of the corresponding polynomial does not intersect the x-axis in the real coordinate plane.

The nature of the roots (real, complex, distinct, equal) is determined by the discriminant, $b^2 - 4ac$, which is a key component of the quadratic formula.

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Terms Related to Quadratic Equations

When working with the standard form of a quadratic equation, $ax^2 + bx + c = 0$, it's useful to be familiar with the terms describing its components:

• Quadratic Term: This is the term containing the variable raised to the power of two, which is $ax^2$.
• Linear Term: This is the term containing the variable raised to the power of one, which is $bx$.
• Constant Term: This is the term that does not contain the variable, which is $c$.
• Coefficient $a$: The numerical factor multiplying the quadratic term ($x^2$). It is the leading coefficient and must be non-zero ($a \neq 0$).
• Coefficient $b$: The numerical factor multiplying the linear term ($x$). It can be any real number, including zero. If $b=0$, the equation is called a "pure quadratic equation" (e.g., $ax^2 + c = 0$).
• Coefficient $c$: The numerical value of the constant term. It can be any real number, including zero. If $c=0$, the equation is of the form $ax^2 + bx = 0$, which can be easily solved by factoring out $x$.

Understanding these terms and the standard form is essential for applying the various methods used to solve quadratic equations and for analysing their properties.

Relation Between the Roots and Coefficients of a Quadratic Equation

Vieta's Formulas for Quadratics

There is a fundamental and very useful connection between the roots (solutions) of a polynomial equation and its coefficients. For quadratic equations, these relationships are straightforward and are a specific instance of what are known as Vieta's formulas. These formulas allow us to determine the sum and product of the roots of a quadratic equation directly from its coefficients, without actually having to solve the equation first.

Consider a quadratic equation in standard form:

Let the two roots of this quadratic equation be $\alpha$ and $\beta$. These roots can be real or complex.

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The Relationships

Vieta's formulas for a quadratic equation state the following relationships between the roots $\alpha, \beta$ and the coefficients $a, b, c$:

1. Sum of the Roots:

   The sum of the two roots ($\alpha + \beta$) is equal to the negative of the ratio of the coefficient of the $x$ term ($b$) to the coefficient of the $x^2$ term ($a$).

   $\alpha + \beta = -\frac{b}{a}$

   [Sum of Roots Formula]

   This can be stated in words as: Sum of roots $= -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2}$.

2. Product of the Roots:

   The product of the two roots ($\alpha \times \beta$ or $\alpha\beta$) is equal to the ratio of the constant term ($c$) to the coefficient of the $x^2$ term ($a$).

   $\alpha \times \beta = \frac{c}{a}$

   [Product of Roots Formula]

   This can be stated in words as: Product of roots $= \frac{\text{Constant term}}{\text{Coefficient of } x^2}$.

These two relationships are incredibly useful for analyzing quadratic equations and their roots.

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Derivation of the Relationships (from factored form)

The relationships between the roots and coefficients can be derived by considering the factored form of the quadratic polynomial. If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$, then by the Factor Theorem, $(x-\alpha)$ and $(x-\beta)$ are linear factors of the corresponding polynomial $P(x) = ax^2 + bx + c$.

Therefore, the polynomial $P(x)$ can be written in its factored form as $P(x) = k(x-\alpha)(x-\beta)$ for some constant $k$. Since $a$ is the coefficient of the $x^2$ term in the standard form $ax^2 + bx + c$, and the $x^2$ term in the expanded factored form is $k \times x \times x = kx^2$, the constant $k$ must be equal to the leading coefficient $a$.

So, we have the identity:

$$ ax^2 + bx + c = a(x-\alpha)(x-\beta) $$

Now, expand the right-hand side (RHS) of the equation:

$$ a(x-\alpha)(x-\beta) = a(x \times x - x \times \beta - \alpha \times x + \alpha \times \beta) $$ $$ = a(x^2 - \beta x - \alpha x + \alpha \beta) $$

Group the terms containing $x$:

$$ = a(x^2 - (\alpha + \beta)x + \alpha \beta) $$

Distribute the coefficient $a$ to each term inside the parentheses:

$$ = ax^2 - a(\alpha + \beta)x + a(\alpha \beta) $$

Now, we compare this expanded form with the standard form of the quadratic polynomial, $ax^2 + bx + c$:

$$ ax^2 + bx + c = ax^2 - a(\alpha + \beta)x + a(\alpha \beta) $$

For two polynomials to be equal for all values of $x$, their corresponding coefficients must be equal. Comparing the coefficients of $x^2$, $x$, and the constant term on both sides:

• Comparing coefficients of $x^2$: $a = a$. This is consistent.
• Comparing coefficients of $x$: $b = -a(\alpha + \beta)$. To isolate the sum of the roots $(\alpha + \beta)$, divide both sides by $-a$ (which is non-zero since $a \neq 0$):

  $\frac{b}{-a} = \alpha + \beta \implies \alpha + \beta = -\frac{b}{a}$

  This matches the formula for the sum of the roots.

• Comparing constant terms: $c = a(\alpha \beta)$. To isolate the product of the roots $(\alpha \beta)$, divide both sides by $a$ (since $a \neq 0$):

  $\frac{c}{a} = \alpha \beta \implies \alpha \beta = \frac{c}{a}$

  This matches the formula for the product of the roots.

This derivation confirms the relationships between the roots and coefficients of a quadratic equation.

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Applications of the Relationships

The relationships between the roots and coefficients of a quadratic equation are very useful in various situations:

• Finding the Sum and Product of Roots: You can quickly find the sum and product of the roots of a quadratic equation by simply looking at its coefficients, without needing to solve the equation first. This is particularly useful when you don't need the individual root values.
• Forming a Quadratic Equation: If you know the roots of a quadratic equation ($\alpha$ and $\beta$), you can easily form the equation. Since $\alpha + \beta = -b/a$ and $\alpha \beta = c/a$, a quadratic equation with these roots can be written as $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$. Multiplying this equation by any non-zero constant $a'$ gives another quadratic equation with the same roots: $a'(x^2 - (\alpha+\beta)x + \alpha\beta) = 0$.
• Checking Solutions: After finding the roots of a quadratic equation using another method (like factoring or the quadratic formula), you can use the sum and product formulas to quickly check if your roots are correct. Sum your found roots and compare with $-b/a$. Multiply your found roots and compare with $c/a$.
• Solving Problems Involving Symmetric Functions of Roots: In some problems, you might be asked to find expressions involving the roots (like $\alpha^2 + \beta^2$ or $\frac{1}{\alpha} + \frac{1}{\beta}$) without finding the roots themselves. These expressions can often be rewritten in terms of the sum ($\alpha+\beta$) and product ($\alpha\beta$), which you can find from the coefficients. For example, $\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$.

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The relationships between the roots and coefficients of a quadratic equation are fundamental tools, offering alternative ways to analyse, construct, and verify quadratic equations and their solutions.

Solving a Quadratic Equation by Factorisation

Using Factoring to Find Roots

The factorisation method is one of the primary algebraic techniques for finding the roots (or solutions) of a quadratic equation $ax^2 + bx + c = 0$. This method relies on the ability to express the quadratic polynomial $ax^2 + bx + c$ as a product of its linear factors. Once the quadratic is factored, we use the powerful Zero Product Property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero. If $A \times B = 0$, then either $A=0$ or $B=0$ (or both).

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Procedure for Solving by Factorisation

To solve a quadratic equation $ax^2 + bx + c = 0$ using the factorisation method:

1. Standard Form:

   Ensure the given quadratic equation is written in the standard form $ax^2 + bx + c = 0$. This means all terms are on one side of the equals sign, arranged in decreasing order of powers of the variable, and the other side is $0$.

2. Factorise the Quadratic Polynomial:

   Factorise the quadratic expression $ax^2 + bx + c$ into a product of its linear factors.
   • If $a=1$ (i.e., the equation is $x^2 + bx + c = 0$), find two numbers $p$ and $q$ such that their sum is the coefficient of $x$ ($p+q=b$) and their product is the constant term ($pq=c$). The factored form is $(x+p)(x+q)$.
   • If $a \neq 1$, find two numbers $p$ and $q$ such that their sum is the coefficient of $x$ ($p+q=b$) and their product is $a \times c$ ($pq=ac$). Rewrite the middle term $bx$ as the sum of two terms, $px+qx$, resulting in a four-term expression $ax^2 + px + qx + c$. Then, factorise this four-term expression by grouping. This should lead to a product of two linear binomials, $(Ax+B)(Cx+D)$.

   Always look for a common factor among $a, b, c$ first before attempting other factorisation methods.

3. Apply the Zero Product Property:

   Once the quadratic polynomial is factored, set the entire factored expression equal to zero. Since the expression is a product of linear factors equal to zero, set each linear factor equal to zero individually.

4. Solve the Linear Equations:

   Solve each of the resulting linear equations for the variable $x$. The values obtained are the roots of the original quadratic equation.

5. Check the Roots:

   (Optional but recommended) Substitute each root back into the original quadratic equation $ax^2 + bx + c = 0$ to verify that it makes the equation true (LHS = RHS).

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The factorisation method is generally the quickest method for solving quadratic equations *if* the quadratic polynomial can be easily factored. However, not all quadratic polynomials with integer coefficients are factorable into linear factors with integer or rational coefficients. In such cases, other methods like completing the square or using the quadratic formula are necessary.

Solving a Quadratic Equation by Completing the Square

Transforming into a Perfect Square

The method of completing the square is a powerful technique for solving any quadratic equation $ax^2 + bx + c = 0$. It involves algebraically manipulating the equation so that one side becomes a perfect square trinomial (a trinomial that is the result of squaring a binomial, like $(x+k)^2$ or $(x-k)^2$). Once one side is a perfect square equal to a constant, the equation can be easily solved by taking the square root of both sides.

This method is significant not only as a way to solve quadratic equations but also because it is used to derive the quadratic formula, which provides a general solution for any quadratic equation.

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Procedure for Solving by Completing the Square

To solve a quadratic equation $ax^2 + bx + c = 0$, where $a \neq 0$, by completing the square:

1. Make Leading Coefficient 1:

   If the coefficient of $x^2$ ($a$) is not $1$, divide the entire equation by $a$. This ensures that the $x^2$ term has a coefficient of $1$. $$ \frac{ax^2}{a} + \frac{bx}{a} + \frac{c}{a} = \frac{0}{a} $$ $$ x^2 + \frac{b}{a} x + \frac{c}{a} = 0 $$

2. Isolate Variable Terms:

   Move the constant term $\frac{c}{a}$ to the right side of the equation. $$ x^2 + \frac{b}{a} x = -\frac{c}{a} $$

3. Complete the Square:

   Focus on the left side, which is now in the form $x^2 + (\text{coefficient of } x)x$. To make it a perfect square trinomial, add the square of half the coefficient of the $x$ term to both sides of the equation.
   • Coefficient of $x$ is $\frac{b}{a}$.
   • Half of this coefficient is $\frac{1}{2} \times \frac{b}{a} = \frac{b}{2a}$.
   • The square of half the coefficient is $\left(\frac{b}{2a}\right)^2 = \frac{b^2}{(2a)^2} = \frac{b^2}{4a^2}$.

   Add $\frac{b^2}{4a^2}$ to both sides:

   $$ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 $$

4. Factor the Perfect Square:

   The left side of the equation is now a perfect square trinomial of the form $x^2 + 2kx + k^2$ or $x^2 - 2kx + k^2$. Specifically, $x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2$ factors as $\left(x + \frac{b}{2a}\right)^2$. $$ \left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} $$

5. Simplify the Right Side:

   Combine the terms on the right side into a single fraction by finding a common denominator, which is $4a^2$. $$ -\frac{c}{a} + \frac{b^2}{4a^2} = -\frac{c}{a} \times \frac{4a}{4a} + \frac{b^2}{4a^2} = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} = \frac{b^2 - 4ac}{4a^2} $$

   So the equation becomes:

   $$ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} $$

6. Take the Square Root:

   Take the square root of both sides of the equation. Remember that taking the square root introduces both a positive and a negative possibility on the right side. $$ \sqrt{\left(x + \frac{b}{2a}\right)^2} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} $$ $$ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{\sqrt{4a^2}} $$

   Simplify the denominator of the square root on the right side: $\sqrt{4a^2} = \sqrt{4} \times \sqrt{a^2} = 2 \times |a|$. However, the $\pm$ sign on the right side already accounts for the sign of $a$, so we can simplify $\sqrt{4a^2}$ as $2a$ in the denominator combined with the $\pm$ symbol.

   $$ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} $$

7. Isolate $x$:

   Subtract $\frac{b}{2a}$ from both sides of the equation to solve for $x$. $$ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} $$

8. Combine Terms:

   Since the terms on the right side have a common denominator ($2a$), they can be combined into a single fraction. $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

The result obtained in Step 8 is the Quadratic Formula. Solving a specific quadratic equation by completing the square involves applying these steps to that equation's coefficients, resulting in the roots.

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The method of completing the square is a systematic way to solve any quadratic equation and is crucial for understanding the origin of the quadratic formula. While it can involve fractions, it always yields the exact roots, whether they are real or complex.

Solving a Quadratic Equation by Using the Quadratic Formula

A General Solution

While factoring and completing the square are valid methods for solving quadratic equations, they can sometimes be cumbersome or difficult to apply directly, especially when coefficients are large or result in non-integer values during intermediate steps. The quadratic formula provides a universal method to find the roots of *any* quadratic equation, regardless of the nature of its coefficients or roots.

The quadratic formula is derived by applying the method of completing the square to the standard form of the general quadratic equation, $ax^2 + bx + c = 0$. As shown in the derivation in the previous section, completing the square on $ax^2 + bx + c = 0$ systematically leads to an expression for $x$ in terms of $a, b,$ and $c$.

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The Quadratic Formula

The roots of a quadratic equation $ax^2 + bx + c = 0$, where $a, b,$ and $c$ are real coefficients and $a \neq 0$, are given by the formula:

Key components of the formula:

• $a, b, c$: These are the coefficients of the quadratic equation when it is written in standard form $ax^2 + bx + c = 0$. Ensure the equation is in this form before identifying $a, b, c$.
• $\pm$: This symbol indicates that there are generally two roots. One root is found by using the plus sign before the square root, and the other root is found by using the minus sign. $$ x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \quad \text{and} \quad x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} $$
• $b^2 - 4ac$: This expression under the square root is called the discriminant, often denoted by the Greek letter Delta ($\Delta$) or $D$. The value of the discriminant determines the nature of the roots (real or complex, distinct or equal), which is discussed in the next section.
• $2a$: This is twice the coefficient of the $x^2$ term and forms the denominator of the entire expression. Note that $a \neq 0$.

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Procedure for Solving Using the Quadratic Formula

To solve a quadratic equation $ax^2 + bx + c = 0$ using the quadratic formula, follow these steps:

1. Standard Form:

   Write the given quadratic equation in its standard form $ax^2 + bx + c = 0$. Make sure all terms are on one side and the equation is set to zero.

2. Identify Coefficients:

   Clearly identify the values of the coefficients $a, b,$ and $c$ from the standard form of the equation. Be careful with signs.

3. Substitute into Formula:

   Substitute the values of $a, b,$ and $c$ into the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

4. Evaluate Discriminant:

   Calculate the value of the expression under the square root, $b^2 - 4ac$.

5. Calculate Square Root:

   Find the square root of the value obtained in Step 4. This might be a real number (positive or zero) or an imaginary number (if the value under the root is negative).

6. Calculate the Roots:

   Use the value of the square root from Step 5 to calculate the two roots separately: one using the plus sign ($\frac{-b + \sqrt{\Delta}}{2a}$) and one using the minus sign ($\frac{-b - \sqrt{\Delta}}{2a}$). Simplify the resulting expressions.

7. Check the Roots:

   (Optional but recommended) Substitute each found root back into the original quadratic equation to verify that it makes the equation true.

Examples

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The quadratic formula is a reliable and comprehensive method for solving any quadratic equation. It is particularly valuable when factoring is difficult or not possible with real numbers and when dealing with equations that have complex roots. The value of the discriminant ($b^2-4ac$) plays a crucial role in determining the nature of these roots.

Nature of Roots of a Quadratic Equation (using Discriminant)

The Discriminant ($\Delta$ or $D$)

When solving a quadratic equation $ax^2 + bx + c = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, the expression under the square root, $b^2 - 4ac$, plays a crucial role. This expression is called the discriminant, often denoted by the symbol $\Delta$ (Delta) or simply $D$. The discriminant provides vital information about the characteristics or "nature" of the roots of the quadratic equation without requiring you to calculate the roots themselves.

The value of the discriminant tells us whether the roots are real numbers or complex numbers, and if they are real, whether they are distinct (different) or equal.

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Interpreting the Discriminant

The nature of the roots of the quadratic equation $ax^2 + bx + c = 0$ (where $a, b, c$ are real coefficients and $a \neq 0$) is determined solely by the value of the discriminant $D = b^2 - 4ac$. There are three possible cases for the value of $D$:

1. Case 1: $D > 0$ (Discriminant is positive)

   If the discriminant is a positive number, then the term $\sqrt{D} = \sqrt{b^2 - 4ac}$ is a real number, and it is non-zero. In the quadratic formula $x = \frac{-b \pm \sqrt{D}}{2a}$, we will be adding and subtracting a non-zero real number ($\sqrt{D}$) from $-b$. This results in two distinct values for $x$. Since $-b$ and $\sqrt{D}$ are real numbers (and $2a$ is real), the resulting values for $x$ will also be real numbers.

   Conclusion: If $b^2 - 4ac > 0$, the quadratic equation has two distinct real roots (two different real number solutions).

   Geometrical Interpretation: The graph of the corresponding quadratic polynomial $y = ax^2 + bx + c$ (a parabola) intersects the x-axis at two different points. The x-coordinates of these intersection points are the two distinct real roots.

2. Case 2: $D = 0$ (Discriminant is zero)

   If the discriminant is exactly zero, then the term $\sqrt{D} = \sqrt{0} = 0$. In the quadratic formula $x = \frac{-b \pm \sqrt{D}}{2a}$, the term $\pm \sqrt{0}$ simplifies to $\pm 0$, which is just $0$. The formula becomes $x = \frac{-b \pm 0}{2a} = \frac{-b}{2a}$.

   Since there is no $\pm$ term involving a non-zero value, both possibilities ($+$ and $-$) yield the same result, $\frac{-b}{2a}$. Thus, the quadratic equation has only one distinct real number as a solution. However, because it arose from a quadratic equation, this root is said to have a multiplicity of 2; it is counted as two equal roots.

   Conclusion: If $b^2 - 4ac = 0$, the quadratic equation has one real root with multiplicity 2 (or two equal real roots).

   Geometrical Interpretation: The graph of the corresponding quadratic polynomial $y = ax^2 + bx + c$ (a parabola) touches the x-axis at exactly one point. This point is the vertex of the parabola, and its x-coordinate is the repeated real root.

3. Case 3: $D < 0$ (Discriminant is negative)

   If the discriminant is a negative number, then the term $\sqrt{D} = \sqrt{b^2 - 4ac}$ is the square root of a negative number. In the system of real numbers, the square root of a negative number is not defined as a real number. It results in an imaginary number. Specifically, if $D = -k$ where $k > 0$, then $\sqrt{D} = \sqrt{-k} = \sqrt{-1 \times k} = \sqrt{-1} \times \sqrt{k} = i\sqrt{k}$, where $i$ is the imaginary unit ($i^2 = -1$).

   In the quadratic formula $x = \frac{-b \pm \sqrt{D}}{2a}$, we will be adding and subtracting an imaginary number from $-b$. This results in two roots that are complex numbers. Specifically, if $b$ and $a$ are real, the roots will be a pair of complex conjugates (of the form $p+qi$ and $p-qi$). These roots are not real numbers.

   Conclusion: If $b^2 - 4ac < 0$, the quadratic equation has no real roots (it has two complex conjugate roots).

   Geometrical Interpretation: The graph of the corresponding quadratic polynomial $y = ax^2 + bx + c$ (a parabola) does not intersect or touch the x-axis at all. The parabola lies entirely above the x-axis (if $a>0$) or entirely below the x-axis (if $a<0$).

In summary, the discriminant $D = b^2 - 4ac$ is a powerful tool to determine the nature of the roots of a quadratic equation:

• $D > 0 \implies$ Two distinct real roots
• $D = 0 \implies$ One real root (repeated)
• $D < 0 \implies$ No real roots (Two complex conjugate roots)

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Examples Determining the Nature of Roots

To determine the nature of the roots, we calculate the discriminant using the coefficients $a, b,$ and $c$ from the standard form of the quadratic equation and then interpret its value.

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Using the discriminant is an efficient way to determine the type of solutions a quadratic equation has without needing to solve the entire equation. It provides important information about where the graph of the corresponding quadratic function intersects the x-axis.