Hard SAT Math Questions
Hard SAT math questions are rarely about obscure formulas. They usually combine familiar skills, hide a shortcut, or ask for an expression instead of a single variable.
What this page covers
Skill-focused practice with answers and explanations.
Free practice questions
Try these first, then use the app to build a full adaptive set.
If x + y = 11 and xy = 24, what is the value of x^2 + y^2?
Use (x + y)^2 = x^2 + 2xy + y^2. Since 11^2 = x^2 + y^2 + 48, x^2 + y^2 = 121 - 48 = 73.
The equation x^2 - 6x + c = 0 has exactly one real solution. What is the value of c?
A quadratic has exactly one real solution when the discriminant is 0. Here b^2 - 4ac = 36 - 4c. Set 36 - 4c = 0, so c = 9.
A circle has center (3, -2) and radius 5. Point (3, k) is on the circle, and k is positive. What is k?
The point has the same x-coordinate as the center, so it is directly above or below the center. A radius of 5 means k is 5 units from -2. The positive value is -2 + 5 = 3.
How to practice this topic
A few rules that make review sessions more useful.
- Look for identities like (x + y)^2 when the question asks for an expression.
- For hard quadratics, check the discriminant, factoring, and vertex form.
- Draw coordinate geometry problems. A quick sketch often reveals the shortcut.
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