4.2 Solving Quadratic Equations by Multiple Methods

4.2 Solving Quadratic Equations by Multiple Methods

Quick Summary: A quadratic equation is a second-degree polynomial equation of the form $ax^2 + bx + c = 0$ (where $a \neq 0$). Solving a quadratic equation means finding all values of $x$ (called roots, zeros, or solutions) that satisfy the equation. On the Digital SAT, choosing the fastest method—factoring, square roots, completing the square, or the quadratic formula—saves critical test time.

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The Four Solution Methods Overview

                         +-----------------------------+
                         |      ax^2 + bx + c = 0      |
                         +--------------+--------------+
                                        |
         +------------------------------+------------------------------+
         |                              |                              |
         v                              v                              v
+-----------------+            +-----------------+            +-----------------+
|  1. FACTORING   |            | 2. SQUARE ROOTS |            | 3. QUAD FORMULA |
| Fastest when    |            | Fastest when    |            | Always works;   |
| easily factored |            | b = 0 or        |            | best for messy  |
| over integers   |            | (x - h)^2 = k   |            | radicals        |
+-----------------+            +-----------------+            +-----------------+

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Method 1: Factoring & The Zero-Product Property

The Zero-Product Property states that if the product of two or more algebraic factors equals zero, at least one factor must equal zero:

$A \cdot B = 0 \iff A = 0 \quad \text{or} \quad B = 0$

Step-by-Step Factoring Workflow

1. Zero Out: Move all terms to one side so the equation equals zero: $ax^2 + bx + c = 0$.
2. GCF: Factor out any common monomial greatest common factor.
3. Decompose: Factor the quadratic expression into linear binomials $(p_1 x + q_1)(p_2 x + q_2) = 0$.
4. Solve: Set each linear factor equal to zero and solve for $x$.

Worked Example 1: Solving by Factoring

$2x^2 - 7x - 15 = 0$

• Product $ac = 2 \times (-15) = -30$, Sum $b = -7$. Factors: $-10$ and $+3$.
• Split middle term: $2x^2 - 10x + 3x - 15 = 0$.
• Group: $2x(x - 5) + 3(x - 5) = 0 \implies (2x + 3)(x - 5) = 0$.
• Set factors to zero: $2x + 3 = 0 \implies x = -\frac{3}{2}$ $x - 5 = 0 \implies x = 5$

Roots: $x = -\frac{3}{2}$ and $x = 5$.

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Method 2: Taking Square Roots (Square Root Property)

When a quadratic equation has no linear term ($b = 0$) or is expressed in completed square form $(x - h)^2 = k$:

$(x - h)^2 = k \implies x - h = \pm \sqrt{k} \implies x = h \pm \sqrt{k}$

[!CAUTION] The Missing Plus-Minus Trap: Never forget the $\pm$ symbol when taking the square root of both sides. For example, $x^2 = 25 \implies x = \pm 5$ ($x = 5$ and $x = -5$). Omitting the negative root is one of the most common SAT errors.

Worked Example 2: Square Root Property

$3(x - 4)^2 - 48 = 0$

1. Isolate the squared term: $3(x - 4)^2 = 48 \implies (x - 4)^2 = 16$
2. Take the square root of both sides: $x - 4 = \pm \sqrt{16} = \pm 4$
3. Split into two cases: $x = 4 + 4 = 8 \quad \text{or} \quad x = 4 - 4 = 0$

Roots: $x = 8$ and $x = 0$.

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Method 3: Completing the Square

Completing the square transforms any quadratic $ax^2 + bx + c = 0$ into vertex/squared form $(x - h)^2 = k$. This is especially useful when finding circle equations or rewriting quadratic functions into vertex form.

Protocol for $x^2 + bx + c = 0$ ($a = 1$)

1. Move the constant $c$ to the right side: $x^2 + bx = -c$.
2. Take half the coefficient of $x$, square it, and add it to both sides: $\left(\frac{b}{2}\right)^2$.
3. Factor the left side as a perfect square: $\left(x + \frac{b}{2}\right)^2 = -c + \left(\frac{b}{2}\right)^2$.
4. Apply the square root property.

Worked Example 3: Completing the Square

$x^2 - 8x + 5 = 0$

• Move constant: $x^2 - 8x = -5$
• Add $\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16$ to both sides: $x^2 - 8x + 16 = -5 + 16$ $(x - 4)^2 = 11$
• Take square roots: $x - 4 = \pm \sqrt{11} \implies x = 4 \pm \sqrt{11}$

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Method 4: The Quadratic Formula

For any quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$, the solutions are given by:

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

Worked Example 4: Quadratic Formula

$2x^2 + 5x - 4 = 0$ Here $a = 2, b = 5, c = -4$.

$x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-4)}}{2(2)} = \frac{-5 \pm \sqrt{25 + 32}}{4} = \frac{-5 \pm \sqrt{57}}{4}$

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SAT Power Shortcut: Vieta's Formulas (Sum & Product of Roots)

Many SAT questions ask for "the sum of the solutions" or "the product of the solutions" rather than the individual roots themselves. Solving the entire equation is completely unnecessary!

For any quadratic equation $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$:

$\text{Sum of Roots: } r_1 + r_2 = -\frac{b}{a}$ $\text{Product of Roots: } r_1 \cdot r_2 = \frac{c}{a}$ $\text{Difference of Roots: } |r_1 - r_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}$

Proof of Vieta's Sum Formula:

$r_1 + r_2 = \frac{-b + \sqrt{b^2-4ac}}{2a} + \frac{-b - \sqrt{b^2-4ac}}{2a} = \frac{-2b}{2a} = -\frac{b}{a}$

Worked Example 5: Instant Sum of Roots

Question: What is the sum of the solutions to $4x^2 - 28x + 13 = 0$?

• Traditional Method: Use quadratic formula, obtain $x = \frac{28 \pm \sqrt{576}}{8}$, calculate both roots, then add them. (Takes 45 seconds).
• Vieta's Shortcut: $\text{Sum} = -\frac{b}{a} = -\frac{-28}{4} = \frac{28}{4} = 7$. (Takes 3 seconds).

Worked Example 6: Sum of Squares of Roots

Question: If $r_1$ and $r_2$ are the roots of $x^2 - 6x + 4 = 0$, what is $r_1^2 + r_2^2$?

• Using the algebraic identity: $r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2$.
• Sum of roots: $r_1 + r_2 = -\frac{-6}{1} = 6$.
• Product of roots: $r_1 r_2 = \frac{4}{1} = 4$.
• Therefore: $r_1^2 + r_2^2 = (6)^2 - 2(4) = 36 - 8 = 28$.

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Method Selection Matrix for the Digital SAT

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Digital SAT Desmos Workflow for Quadratic Roots

1. Type the Equation: Enter y = 2x^2 - 7x - 15.
2. Identify $x$-Intercepts: Desmos automatically generates gray dots where the curve crosses the $x$-axis ($y = 0$).
3. Read the Coordinates: Click the gray dots to inspect the roots $(-1.5, 0)$ and $(5, 0)$.
4. Fraction Conversion: In Desmos, type -1.5 on a new line and click the fraction toggle icon to get $-\frac{3}{2}$.
5. Direct Root Equation: Alternatively, type 0 = 2x^2 - 7x - 15 into Desmos. Vertical dashed lines will appear at the exact root locations $x = -1.5$ and $x = 5$.

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Common Pitfalls & Exam Traps

• Dividing by $x$ and losing a root: In $x^2 = 5x$, dividing both sides by $x$ gives $x = 5$, completely losing the solution $x = 0$. Always factor: $x(x - 5) = 0 \implies x = 0, 5$.
• Forgetting $\pm$ when taking square roots: $(x - 3)^2 = 16 \implies x - 3 = \pm 4$, giving $x = 7$ and $x = -1$.
• Sign confusion in Quadratic Formula: If $b = -6$, then $-b = -(-6) = +6$. Also, $(-6)^2 = +36$, never $-36$.
• Vieta's Formula sign: The sum of roots has a negative sign ($-\frac{b}{a}$), while the product of roots does not ($\frac{c}{a}$).