Use the structure of an expression to identify ways to rewrite it, and produce equivalent forms to reveal properties of the quantity (Ohio A-SSE.2, A-SSE.3).

What this topic is asking

Ohio standards A-SSE.2 and A-SSE.3 ask you to rewrite an expression into an equivalent form by reading its structure, then to choose the form that reveals a useful property: a common factor, a difference of squares, the zeros of a quadratic, or a starting value. The skill is recognizing the pattern, not blindly applying one rule. This sits in the Number and Quantity, Expressions and Equations reporting category.

First move: factor out the GCF

Before any other pattern, pull out the greatest common factor of all terms. The GCF includes the largest shared number and the lowest shared power of each variable.

Recognizing a difference of squares

A difference of squares is two perfect squares subtracted. It factors with a fixed pattern.

For $x^2 - 49$, write it as $x^2 - 7^2 = (x - 7)(x + 7)$. Often you factor out a GCF first and then a difference of squares appears: $2x^2 - 50 = 2(x^2 - 25) = 2(x - 5)(x + 5)$.

Choosing the form that reveals a property

A-SSE.3 stresses that the right form depends on the question. The same quadratic can be written three ways, each exposing something different.

• Standard form $ax^2 + bx + c$: the constant $c$ is the $y$-intercept (the value at $x = 0$).
• Factored form $a(x - r_1)(x - r_2)$: the numbers $r_1$ and $r_2$ are the zeros (the $x$-intercepts).
• Vertex form $a(x - h)^2 + k$: the point $(h, k)$ is the vertex (the maximum or minimum).

So if a question asks for the zeros, rewrite into factored form; if it asks for the lowest point, rewrite into vertex form. Recognizing which property you need tells you which rewrite to perform.

How Ohio examines this topic

• Multiple-select. Pick every expression equivalent to a given one; verify candidates by expanding.
• Equation response. Type the factored form (GCF, difference of squares, or trinomial).
• Drag and drop. Match each form (standard, factored, vertex) to the property it reveals.

Because equivalence is exact, the safest check on any rewrite is to expand it back and compare term by term.

Why equivalent forms are interchangeable but not equal in usefulness

Two expressions are equivalent when they produce the same output for every input, so substituting any number gives the same value either way. That guarantees you can replace one with the other without changing the math. What changes is what you can read off. The factored form $(x - 3)(x + 3)$ instantly shows the zeros $x = 3$ and $x = -3$, while the equivalent $x^2 - 9$ hides them but shows the $y$-intercept $-9$ at a glance. Neither is "more correct", but each is the better tool for a different question. This is why the test asks you to choose and produce a specific form: the grader is checking that you can move to the representation that exposes the property in the prompt. Building the habit of asking "which form shows what I need?" before you compute saves time and avoids unnecessary algebra.

A note on always factoring the GCF first

Skipping the GCF is the most common reason a rewrite looks wrong or an exact-match item is marked incorrect. If you try to factor $4x^2 - 36$ as a difference of squares directly, you might write $(2x - 6)(2x + 6)$, which is correct but not fully factored, since each binomial still shares a factor of $2$. Pulling the $4$ out first, $4(x^2 - 9) = 4(x - 3)(x + 3)$, gives the cleaner, fully factored result the test usually wants. Make the GCF your reflex first step.

Try this

Q1. Rewrite $15x^2 + 25x$ in factored form. [1 point]

• Cue. GCF is $5x$, so $5x(3x + 5)$.

Q2. Factor $x^2 - 64$ completely. [1 point]

• Cue. Difference of squares: $(x - 8)(x + 8)$.