Solve quadratic equations (factoring, completing the square, or quadratic formula). | Nonlinear equations in one variable | SAT Math

Quadratics can be solved by factoring, the quadratic formula, or completing the square.

核心知识

For $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 = 0$ , try factoring first. If the numbers don't factor cleanly, use the quadratic formula: $𝑥 = - 𝑏 \pm \sqrt 𝑏 2 - 4 𝑎 𝑐 2 𝑎$ . Completing the square is most useful when the problem asks for vertex form or when $𝑎 = 1$ and $𝑏$ is even.

深入理解

The SAT gives you quadratics in three common forms:

Standard form: $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 = 0$
Factored form: $𝑎 (𝑥 - 𝑟) (𝑥 - 𝑠) = 0$
Vertex form: $𝑎 (𝑥 - ℎ) 2 + 𝑘 = 0$

When the equation is already in standard form, check whether it factors. If $𝑎 = 1$ , you're looking for two numbers that multiply to $𝑐$ and add to $𝑏$ . If it doesn't factor easily, go straight to the quadratic formula.

Completing the square converts $𝑥 2 + 𝑏 𝑥 + 𝑐 = 0$ into $(𝑥 + 𝑏 2) 2 = 𝑏 2 4 - 𝑐$ . This is cleanest when $𝑏$ is even. The SAT sometimes asks you to rewrite a quadratic in vertex form — that's completing the square.

On timed tests, factoring is fastest when it works. The quadratic formula always works but is slower. Pick the method that matches the numbers you see.

分步讲解

Move all terms to one side so the equation equals 0.
Try factoring: look for two numbers that multiply to $𝑎 𝑐$ and add to $𝑏$ .
If factoring doesn't work quickly, apply the quadratic formula with $𝑎$ , $𝑏$ , and $𝑐$ from $𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 = 0$ .
Simplify the radical and reduce the fraction if possible.

常见误解

Forgetting to set the equation equal to 0 before factoring or applying the formula.
Sign errors in the quadratic formula — the formula has $- 𝑏$ , so if $𝑏$ is negative, $- 𝑏$ is positive.
Dividing by $2 𝑎$ but only applying it to part of the numerator. The entire expression $- 𝑏 \pm \sqrt 𝑏 2 - 4 𝑎 𝑐$ is divided by $2 𝑎$ .
Stopping after finding one solution when the equation has two.

题目

示例解析

What are the solutions to $2 𝑥 2 - 7 𝑥 - 15 = 0$ ?

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