Lesson 98: Solving Quadratic Equations by Factoring

New Concept

If the product of two quantities equals zero, at least one of the quantities equals zero.

What’s next

Next, you’ll use this property to factor and find the roots of several quadratic equations, including real-world application problems.

Property

If the product of two quantities equals zero, at least one of the quantities must be zero. If $ab=0$, then either $a=0$ or $b=0$.

Explanation

Think of it this way: the only way to get zero by multiplying is if one of your numbers is zero! This cool rule lets you split one complicated factored equation into two simple ones you can easily solve. It’s like getting a two-for-one deal on solving math problems!

Examples

• If $(y+2)(y-7) = 0$, then $y+2=0$ or $y-7=0$, so the solutions are $y=-2$ or $y=7$.
• For $3a(a-6) = 0$, you set $3a=0$ or $a-6=0$, which gives the roots $a=0$ or $a=6$.

Property

To solve a quadratic equation, write it in standard form, $ax^2 + bx + c = 0$. Then, factor the quadratic expression and use the Zero Product Property to find the roots.

Explanation

First, get all terms to one side so your equation equals zero. Next, factor the polynomial into smaller pieces. Finally, set each piece equal to zero and solve. This three-step method turns a tricky quadratic into simple mini-problems that are way easier to crack.

Examples

• To solve $x^2 + 4x = 12$, first write it as $x^2 + 4x - 12 = 0$. Then factor to get $(x+6)(x-2)=0$. The roots are $x=-6$ and $x=2$.
• To solve $5x^2 - 2 = 3x$, rewrite as $5x^2 - 3x - 2 = 0$. Then factor to get $(5x+2)(x-1)=0$. The roots are $x=-\frac{2}{5}$ and $x=1$.

Let's tackle a quadratic equation that isn't in standard form yet. This example shows one of the most important key ideas from this lesson: Solving Quadratic Equations by Factoring.

Example Problem

Find the roots of the equation $3x^2 - 14x = 5$.

Step-by-Step

1. First, we need to set the equation equal to 0 by writing it in the standard form $ax^2 + bx + c = 0$.

2. Next, we factor the quadratic expression on the left side.

3. Now we apply the Zero Product Property. If the product of these two factors is zero, then at least one of the factors must be zero. We set each factor equal to zero.

4. Finally, we solve each of these linear equations for $x$.

5. Check your answers by substituting them back into the original equation.

The roots are $-\frac{1}{3}$ and $5$.

Property

Before factoring a quadratic, always check if a Greatest Common Factor (GCF) can be factored out from all terms. This simplifies the equation, making it much easier to solve.

Explanation

Always look for a GCF first! Pulling out the largest number or variable that every term shares makes the rest of the problem simpler. This awesome first step makes the remaining polynomial smaller, cleaner, and a lot less intimidating to factor and solve. It’s a game-changer!

Examples

• To solve $3x^2 - 24x + 48 = 0$, first factor out the GCF of 3 to get $3(x^2 - 8x + 16) = 0$. Then factor to $3(x-4)(x-4)=0$. The only root is $x=4$.
• To solve $12x^2 = 27x$, rewrite as $12x^2 - 27x = 0$. Factor out the GCF of $3x$ to get $3x(4x - 9) = 0$. The roots are $x=0$ and $x=\frac{9}{4}$.