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Math

Quadratic Equations

4 min readMedium10-question skill check

A quadratic equation is any equation where the highest power of $x$ is squared. They show up wherever you're solving for a moment in time, a width and length, or where a parabola crosses zero — and the SAT tests them in three reliable patterns.

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Quadratic Equations

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A quadratic equation is one that can be written in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are numbers and $a \neq 0$ . Examples: $x^2 - 7x + 10 = 0$ , $2x^2 + 5x - 3 = 0$ , $x^2 = 16$ .

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Graph of f(x) = x² − 7x + 10. The roots (where the parabola crosses zero) are x = 2 and x = 5.

There are three ways to solve a quadratic on the SAT, and you should know all three:

1. Factoring (fastest when it works). Find two numbers that multiply to $c$ and add to $b$ . Rewrite the equation as $(x - p)(x - q) = 0$ , then set each factor to zero.

For $x^2 - 7x + 10 = 0$ : find numbers that multiply to $10$ and add to $-7$ . Those are $-2$ and $-5$ . So $(x - 2)(x - 5) = 0$ , giving solutions $x = 2$ and $x = 5$ .

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Both roots are positive — questions like "the larger positive solution" mean 5, not 2.

2. The quadratic formula (always works).

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression under the square root, $b^2 - 4ac$ , is the discriminant. It tells you how many real solutions exist:

Positive: two distinct solutions.
Zero: one repeated solution.
Negative: no real solutions (the parabola never crosses zero).

3. Square-rooting both sides (special cases). If the equation is just $x^2 = 16$ , take the square root of both sides: $x = \pm 4$ . Don't forget the negative root — that trips students often.

A few related ideas:

The graph of a quadratic is a parabola — a U-shape that opens up if $a > 0$ , down if $a < 0$ .
The x-intercepts of the parabola are the solutions to the equation when set to zero.
The vertex (highest or lowest point) is at $x = -b / (2a)$ .

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Quick check

Pause and check yourself before the harder examples. Try factoring first; if it doesn't factor cleanly, fall back to the quadratic formula.

If x² - 11x + 24 = 0, what is the sum of the solutions?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

What are the solutions to $x^2 - 12x + 32 = 0$ ?

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Example 2

What is the larger positive solution to $x^2 - 7x + 10 = 0$ ?

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Example 3

Solve $2x^2 + 3x - 5 = 0$ .

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Common pitfalls

Forgetting the $\pm$ when taking square roots

$x^2 = 16$ has TWO solutions: $x = 4$ and $x = -4$ . Students who write only $x = 4$ miss the negative root. Whenever you square-root both sides, both signs are possible.

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Sign errors when factoring

For $x^2 - 7x + 10 = 0$ , the answers are $x = 2$ and $x = 5$ (both positive). The factors are $(x - 2)(x - 5)$ — minus signs. Students often leave $(x + 2)(x + 5)$ and get $x = -2, -5$ (wrong direction). Always FOIL after factoring to verify.

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Misreading 'the positive solution' when both are positive

Read the question literally: the positive solution? the larger? the smaller? Always check whether your two roots are both positive, both negative, or one of each.

Discriminant arithmetic errors

For $b^2 - 4ac$ , square $b$ first, then subtract $4ac$ . A common slip is forgetting that $4ac$ includes the sign of $c$ . If $c$ is negative, $4ac$ is negative, and you're subtracting a negative — adding.

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Key takeaways

Standard form: $ax^2 + bx + c = 0$ . Factor when you can; use the formula when you can't.
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Factoring shortcut: find two numbers that multiply to $c$ and add to $b$ .
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Quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Always works.
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The discriminant $b^2 - 4ac$ tells you how many real solutions exist.
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Read what the question asks for: both, the larger, the positive, etc.

Skill check

10 questions on Nonlinear Equations in 1 Variable and Systems of Equations in 2 Variables, from our question bank. Score 70% or better across 10 answers to reach Silver and pass this step.

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Where this lesson fits

Before

Nonlinear Functions→

Quadratic Equations

You are here

→Ratios, Rates, Proportional Relationships, and Units

Worked examples

Common pitfalls

Key takeaways

Further reading

Skill check