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.
Graph of f(x) = x² − 7x + 10. The roots (where the parabola crosses zero) are x = 2 and x = 5.
Loading…
Both roots are positive — questions like "the larger positive solution" mean 5, not 2.
Loading…
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?
Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).
Worked examples
Example 1
Loading…
Example 2
Loading…
Example 3
Loading…
Common pitfalls
Forgetting the $\pm$ when taking square roots
Loading…
Sign errors when factoring
Loading…
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
Loading…
Key takeaways
Loading…
Loading…
Loading…
Loading…
Read what the question asks for: both, the larger, the positive, etc.
A quadratic equation is one that can be written in the form ax2+bx+c=0, where a, b, and c are numbers and a=0. Examples: x2−7x+10=0, 2x2+5x−3=0, x2=16.
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 x2−7x+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.
2. The quadratic formula (always works).
x=2a−b±b2−4ac
The expression under the square root, b2−4ac, 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 x2=16, take the square root of both sides: x=±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).
What are the solutions to x2−12x+32=0?
What is the larger positive solution to x2−7x+10=0?
Solve 2x2+3x−5=0.
x2=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.
For x2−7x+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.
For b2−4ac, 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.
Standard form: ax2+bx+c=0. Factor when you can; use the formula when you can't.
Factoring shortcut: find two numbers that multiply to c and add to b.
Quadratic formula: x=2a−b±b2−4ac. Always works.
The discriminant b2−4ac tells you how many real solutions exist.