Solve quadratic equations by factoring and applying the zero-product property, after writing the equation in standard form equal to zero (Ohio A-REI.4b, A-SSE.3a).

What this topic is asking

Ohio standard A-REI.4b asks you to solve quadratic equations, and the first method is factoring with the zero-product property. Write the equation in standard form $ax^2 + bx + c = 0$, factor the left side, and set each factor to zero. The solutions are the zeros (roots, $x$-intercepts) of the related parabola. This is a core skill in the Quadratics block and appears on Part 1, where no calculator helps.

Step 1: standard form, then factor

The zero-product property only works when one side is zero, so rearrange first.

The zero-product property

This single rule is what turns a factored quadratic into solutions.

So $(x - 2)(x - 5) = 0$ splits into $x = 2$ or $x = 5$, but $(x - 2)(x - 5) = 4$ does not split, the right side is not zero.

Watch the GCF case

When the constant term is zero, factor out the common $x$ rather than dividing it away.

For $2x^2 - 10x = 0$, factor $2x(x - 5) = 0$, giving $x = 0$ or $x = 5$. Dividing both sides by $2x$ would lose $x = 0$.

How Ohio examines this topic

• Equation response. Type both solutions (the test usually wants both roots).
• Multiple choice and multiple-select. Pick the solution set, with "lost a root" and "wrong sign" distractors.
• Drag and drop. Match an equation to its solutions or its factors.

Why a quadratic has two solutions

A factored quadratic $(x - p)(x - q) = 0$ is a product of two linear factors, and the zero-product property says the product vanishes exactly when one of the factors does, at $x = p$ or $x = q$. That is why a quadratic typically has two solutions: two factors, two chances to make the product zero. Graphically, these are the two places the parabola crosses the $x$-axis. Sometimes the two factors are identical, $(x - 3)^2 = 0$, giving a single repeated root where the parabola just touches the axis, and sometimes the quadratic does not factor over the reals, in which case it has no real solutions, the parabola misses the axis. Recognizing "two factors, two roots" frames why solving means setting each factor to zero.

Why dividing by the variable loses a root

The most damaging shortcut in this topic is dividing both sides by $x$. Division is only valid by a nonzero quantity, but $x = 0$ may well be a solution, so dividing by $x$ silently assumes $x \neq 0$ and throws that root away. For $x^2 = 6x$, dividing gives $x = 6$ and hides $x = 0$, half the answer gone. Factoring out the $x$ instead, $x(x - 6) = 0$, treats both factors on equal footing and keeps every root. The rule is simple: never divide an equation by an expression that could be zero; factor it out and apply the zero-product property.

Try this

Q1. Solve $x^2 - 9 = 0$ by factoring. [2 points]

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

Q2. Solve $x^2 + 5x = 0$. [2 points]

• Cue. $x(x + 5) = 0$, so $x = 0$ or $x = -5$.