Rewrite expressions to reveal structure (e.g., factors/zeros). | Equivalent expressions | SAT Math

Sometimes the SAT asks you to rewrite an expression in a specific form—vertex form, factored form, or a form that isolates a particular value. The goal is to make a hidden property visible.

Core Idea

Rewriting an expression into an equivalent form can reveal information that's hidden in the original: completing the square exposes the vertex, factoring exposes the zeros, and isolating terms exposes rates or constants.

Understanding

These questions give you an expression and a target form, then ask for a specific constant or feature. Your job is not just to simplify - it is to reshape the expression so a particular piece of information becomes readable.

Common targets:

Vertex form to expose the vertex
Factored form to expose the zeros
Isolated form to expose a rate or constant

Completing the square is the most tested technique here. To convert $𝑥 2 + 𝑏 𝑥 + 𝑐$ into $(𝑥 - ℎ) 2 + 𝑘$ : take half of $𝑏$ , square it, then add and subtract that value inside the expression. Group the perfect square trinomial and simplify the leftover constant.

If the leading coefficient is not 1, factor it out from the $𝑥 2$ and $𝑥$ terms before completing the square. This extra step is where most errors occur - the factored-out coefficient changes the constant you add and subtract.

Factoring to reveal zeros works the same way: rewrite $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐$ as $𝑎 (𝑥 - 𝑟) (𝑥 - 𝑠)$ so the zeros $𝑟$ and $𝑠$ are directly readable.

Step by Step

Identify the target form the question asks for (vertex form, factored form, or other).
If vertex form: factor out the leading coefficient from the quadratic and linear terms, then complete the square.
If factored form: factor using GCF, difference of squares, or trinomial methods.
Match your rewritten expression to the target form and read off the requested value.

Misconceptions

Forgetting to factor out the leading coefficient before completing the square, which throws off the constant term.
Adding a value to complete the square without subtracting it back, changing the expression instead of rewriting it equivalently.
Confusing the vertex form signs: in $(𝑥 - ℎ) 2 + 𝑘$ , the vertex is at $𝑥 = ℎ$ , not $𝑥 = - ℎ$ .

Question

Worked Example

The expression $𝑥 2 - 10 𝑥 + 30$ is equivalent to $(𝑥 - 𝑝) 2 + 𝑞$ . What is the value of $𝑞$ ?

Select an answer to see the explanation