These are 5 hard SAT math questions from our question bank. Every one is at the top of our difficulty scale, the kind you see late in a harder second module. Try each one before you open the solution. Each solution shows the fastest path and links to the notes for that skill.
If you get 4 or 5 right without help, you are working at the level of the hardest Math questions. If you miss more than 2, review the linked notes before you try more questions at this level.
Question 1 (Hard)
In the expression below, a is an integer.
12x2 + ax − 20
If 3x + 4 is a factor of the expression above, what is the value of a ?
- A. 1
- B. 2
- C. 3
- D. 4
Show answer and solution
Answer: A. 1
If 3x + 4 is a factor, then x = −4/3 makes the expression zero. 12(16/9) + a(−4/3) − 20 = 0 gives 64/3 − 4a/3 − 20 = 0. Multiply by 3: 64 − 4a − 60 = 0, so a = 1.
Tip: Plug in the root instead of trying to factor. It works for any “is a factor” question.
Review: Quadratic Equations notes
Question 2 (Hard)
- A. 4
- B. 6
- C. 8
- D. 10
Show answer and solution
Answer: D. 10
Square both sides: (A + B)2 = 2(AB + 5), so A2 + 2AB + B2 = 2AB + 10. The 2AB terms cancel, leaving A2 + B2 = 10.
Tip: You never need A or B. When a question asks for a combination, look for an identity that produces it.
Review: Exponents and Radicals notes
Question 3 (Hard)
2x + ky = 22
In the given system of equations, m and k are constants. The graphs of these equations in the xy-plane intersect at the point (5, y). What is the value of m?
- A. -19
- B. 20
- C. 19
- D. -20
Show answer and solution
Answer: A. -19
Plug x = 5 into the second equation: 10 + ky = 22, so ky = 12. Then the first equation becomes 5m + 12 = −83, so 5m = −95 and m = −19.
Tip: Treat ky as one unknown. You do not need k or y separately.
Review: Systems of Equations notes
Question 4 (Hard)
The function g is a quadratic function. In the xy-plane, the graph of y = g(x) has a vertex at (-1, -4) and passes through the points
(-2, -43) and (1, -160).
What is the value of g(0) - g(2)?
- A. -121
- B. 0
- C. 117
- D. 312
Show answer and solution
Answer: D. 312
Write g(x) = a(x + 1)2 − 4. Use (−2, −43): a − 4 = −43, so a = −39. Check with (1, −160): −39(4) − 4 = −160. Then g(0) = −39 − 4 = −43 and g(2) = −39(9) − 4 = −355. g(0) − g(2) = −43 + 355 = 312.
Tip: Start from vertex form whenever the question gives you the vertex.
Review: Quadratic Functions notes
Question 5 (Hard)
(x + 3)2 + (y - 4)2 = 25
In the xy-plane, the graph of the given equation is a circle. Which point lies on this circle?
- A. (-3, 4)
- B. (3, -4)
- C. (√11 + 3, √14 - 4)
- D. (√11 - 3, √14 + 4)
Show answer and solution
Answer: D. (√11 - 3, √14 + 4)
Test choice D in the equation: x + 3 = √11 and y − 4 = √14. Then (√11)2 + (√14)2 = 11 + 14 = 25. Choices A and B fail because (−3, 4) is the center, and C has the signs flipped.
Tip: For “which point lies on the circle,” substitute each choice. The center is a common trap.
Review: Circles notes
Want more questions at this level?
Targeted Practice builds sets from the skills you miss.
How to get better at hard SAT math
- Look for the shortcut first. Most of these have a short path, like plugging in a root, squaring both sides, or using symmetry. Hard questions reward the idea, not long algebra.
- Learn each skill from the notes linked in the solutions, then drill it in Targeted Practice.
- Watch it explained. The free Advanced Math video series with Mr. O walks through the Advanced Math domain.
- Test yourself under time with a full-length test in the question bank.
Want 5 more? Try the hard SAT math practice test.
Frequently asked questions
What makes an SAT math question hard?
Usually one extra step or one trap, not harder math. Most hard questions use Algebra and Advanced Math skills you already know, with a twist like an unknown constant or a hidden identity.
How many math questions are on the SAT?
44, in two modules of 22 questions and 35 minutes each. How you do on Module 1 decides whether Module 2 is harder or easier.
Can I use a calculator?
Yes, on every Math question. Desmos is built into Bluebook.
Sources
- College Board, How the SAT Is Structured.
- College Board, The Math Section: Overview.
Frequently Asked Questions
What makes an SAT math question hard?
Usually one extra step or one trap, not harder math.
How many math questions are on the SAT?
44, in two modules of 22 questions and 35 minutes each.
Can I use a calculator?
Yes, on every Math question. Desmos is built into Bluebook.