Advanced Math • 35% of test
Quadratic Equations
Select your difficulty level to start practicing. We recommend mastering each level before moving to the next.
Beginner5 Sets
Beginner Practice
Start here to build your foundation. Clear problems and straightforward calculations.
Target Score
400-550
Intermediate5 Sets
Intermediate Practice
Level up with more complex equations and multi-step problems.
Target Score
550-700
Advanced3 Sets
Advanced Practice
Master the hardest concepts. Complex word problems and abstract applications.
Target Score
700-800
What is Quadratic Equations?
Quadratic equations (ax² + bx + c = 0) are second only to linear equations in frequency. The SAT tests factoring, the quadratic formula, completing the square, and interpreting parabola properties.
Step-by-Step Strategy
⚠️ Common Traps to Avoid
Frequently Asked Questions
When should I use the quadratic formula?
When the equation doesn't factor easily, or when you need exact answers. Always works!
What does the discriminant tell me?
b²-4ac > 0: two real solutions. = 0: one real solution. < 0: no real solutions.
How do I factor a quadratic?
Find two numbers that multiply to c and add to b. Factor as (x + p)(x + q) where p×q = c and p+q = b.
What's the vertex of a parabola?
The highest or lowest point. X-coordinate = -b/2a. Substitute to find y-coordinate.
Can Desmos solve quadratics?
Yes! Enter y = ax² + bx + c and find where it crosses the x-axis. Using the <a href='/math'>calculators in our Math practice section</a> is a key SAT strategy.
What are roots, solutions, and x-intercepts?
They all mean the same thing: the x-values where the quadratic equals zero.
What is the relationship between the vertex and the roots?
The x-coordinate of the vertex (-b/2a) is always exactly halfway between the two roots of the quadratic.
How do I find the maximum or minimum value of a quadratic?
The max/min occurs at the vertex. Calculate the y-coordinate by substituting x = -b/2a into the original function.
What if the discriminant is a perfect square?
That means the quadratic can be factored over rational numbers and the roots will be integers or simple fractions.
How do I 'Complete the Square' quickly?
1) Ensure a=1, 2) Take half of b, square it, and add/subtract it to create a perfect square trinomial (x + b/2)².
What is the 'Sum and Product of Roots' rule?
For ax² + bx + c = 0, the sum of the roots is -b/a and the product is c/a. This is a massive time-saver for 'sum of solutions' questions.
How do I interpret the 'a' coefficient in context?
If a > 0, the parabola opens upward (minimum). If a < 0, it opens downward (maximum). Larger |a| means a 'narrower' parabola.
What does 'Complex Roots' mean on the SAT?
It means the discriminant is negative and the parabola never touches the x-axis. Solutions involve 'i', the imaginary unit.
Can I find the intersection of a line and a parabola?
Yes, set the linear equation equal to the quadratic (mx + b = ax² + bx + c) and solve the resulting quadratic for x.
Is factoring always the best method?
No. If the numbers are large, Desmos or the quadratic formula is faster. Master factoring basics in our <a href='/math/quadratic-equations/beginner'>Beginner Quadratic sets</a>.