Solve quadratic equations by factoring using the zero-product property, after writing the equation in standard form (A.PAR, Patterning and Algebraic Reasoning).

What this topic is asking

This Patterning and Algebraic Reasoning (A.PAR) standard asks you to solve quadratic equations by factoring, the fastest method when the quadratic factors with integers. The engine is the zero-product property, which only works when a product equals zero, so the first move is always to write the equation in standard form equal to zero. The Georgia Milestones EOC tests factoring as numeric-entry items (solve and type the solutions) and as multiple-choice items, and it connects directly to graphing: the solutions are the x-intercepts (zeros) of the parabola.

Set it equal to zero first

The zero-product property is the reason factoring solves equations, and it requires a product equal to zero.

So $x^2 + 5x = 24$ must become $x^2 + 5x - 24 = 0$ first. Factoring the left side while the right side is 24 tells you nothing, because the property needs zero, not 24.

Factor and split

Once in standard form, factor the quadratic (see the polynomial-operations page for the factoring methods), then set each factor to zero.

Each solution is the opposite sign of the constant in its factor: the factor $(x + 5)$ gives the solution $x = -5$.

Solutions are the zeros of the parabola

The solutions of $ax^2 + bx + c = 0$ are exactly the x-intercepts of the graph $y = ax^2 + bx + c$, the points where the parabola crosses the x-axis (also called the zeros or roots). So solving by factoring and reading x-intercepts off a graph are two views of the same thing. For $x^2 + 2x - 15 = 0$, the parabola crosses at $(-5, 0)$ and $(3, 0)$.

How the Milestones examines this topic

• Numeric entry. Solve a factorable quadratic and type both solutions.
• Multiple choice. Choose the solutions, or identify the first step (set to zero), with sign-reversal distractors.
• Hot spot. Click the x-intercepts of a parabola whose equation factors.

Why the solutions have opposite signs from the factors

A frequent slip is reading the solution of $(x + 5)$ as $+5$ instead of $-5$, so it helps to see why the sign flips. The factor $(x + 5)$ equals zero when $x + 5 = 0$, which means $x = -5$. You are solving for the value that makes the factor vanish, and that value is the opposite of the constant added inside. So $(x - 3)$ gives $x = +3$ and $(x + 5)$ gives $x = -5$. Saying "the solution is what makes this factor zero" out loud as you split keeps the signs correct, and it also reinforces the zero-product logic, since the whole point is to find the inputs that make a factor, and therefore the product, equal to zero.

When a GCF or special pattern helps

Factoring a quadratic to solve it uses the same toolkit as the polynomial strand. Pull out a GCF first if there is one: $2x^2 - 8x = 0$ becomes $2x(x - 4) = 0$, giving $x = 0$ or $x = 4$. Recognize a difference of squares: $x^2 - 9 = 0$ factors as $(x - 3)(x + 3) = 0$, so $x = \pm 3$. And note that an equation like $x^2 - 8x = 0$ has $x = 0$ as one solution, which students sometimes lose by dividing both sides by $x$ (never divide by a variable, because you discard the $x = 0$ root). Bringing the GCF and difference-of-squares patterns into the solving step is what makes factoring quick on the EOC.

Try this

Q1. Solve $x^2 - 7x + 12 = 0$. [1 point]

• Cue. $(x - 3)(x - 4) = 0$, so $x = 3$ or $x = 4$.

Q2. Solve $x^2 + 4x = 0$. [1 point]

• Cue. $x(x + 4) = 0$, so $x = 0$ or $x = -4$ (do not divide by $x$).