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.
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.
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.
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.
Read what the question asks for: both, the larger, the positive, etc.