The rules that solve most questions
- Factor x² + bx + c by finding two numbers that multiply to c and add to b.
- If (x − p)(x − q) = 0, the solutions are x = p and x = q.
- For ax² + bx + c = 0, the solutions add to −b/a and multiply to c/a.
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. The discriminant b² − 4ac tells you how many real solutions there are (positive: two; zero: one; negative: none).
Traps the wrong answers are built on
- Taking the signs from the factors instead of the solutions: (x + 7) = 0 means x = −7.
- Answering b instead of −b for the sum of solutions.
Worked example
x² − 9x + 20 = 0
What is the sum of the solutions to the equation above?
- A. 20
- B. −9
- C. 9
- D. 1
Answer: C. Factor: find two numbers that multiply to 20 and add to −9. They are −4 and −5, so the equation is (x − 4)(x − 5) = 0. The solutions are x = 4 and x = 5. Their sum is 9. (Shortcut: for x² + bx + c = 0 the sum of the solutions is −b.)
Practice set
Answer each question to see whether you're right and why. When you finish, draw a new set: the numbers change every time.
0 of 10 answered · 0 correct · set 1
- Question 1Medium
x² − 9x + 20 = 0
What is the sum of the solutions to the equation above?
Show answer and explanation
Answer: C
Factor: find two numbers that multiply to 20 and add to −9. They are −4 and −5, so the equation is (x − 4)(x − 5) = 0.
The solutions are x = 4 and x = 5.
Their sum is 9. (Shortcut: for x² + bx + c = 0 the sum of the solutions is −b.)
- Question 2Medium
x² − 15x + 56 = 0
What is the sum of the solutions to the equation above?
Show answer and explanation
Answer: C
Factor: find two numbers that multiply to 56 and add to −15. They are −7 and −8, so the equation is (x − 7)(x − 8) = 0.
The solutions are x = 7 and x = 8.
Their sum is 15. (Shortcut: for x² + bx + c = 0 the sum of the solutions is −b.)
- Question 3Medium
x² − 2x − 63 = 0
What is the positive solution to the equation above?
Show answer and explanation
Answer: A
Factor: find two numbers that multiply to −63 and add to −2. They are −9 and 7, so the equation is (x − 9)(x + 7) = 0.
The solutions are x = 9 and x = −7.
The positive solution is 9.
- Question 4Medium
x² + 4x − 32 = 0
What is the positive solution to the equation above?
Show answer and explanation
Answer: A
Factor: find two numbers that multiply to −32 and add to 4. They are 8 and −4, so the equation is (x + 8)(x − 4) = 0.
The solutions are x = −8 and x = 4.
The positive solution is 4.
- Question 5Medium
x² − 4x − 21 = 0
What is the sum of the solutions to the equation above?
Show answer and explanation
Answer: D
Factor: find two numbers that multiply to −21 and add to −4. They are 3 and −7, so the equation is (x + 3)(x − 7) = 0.
The solutions are x = −3 and x = 7.
Their sum is 4. (Shortcut: for x² + bx + c = 0 the sum of the solutions is −b.)
- Question 6Medium
x² + 3x + 0 = 0
What is the sum of the solutions to the equation above?
Show answer and explanation
Answer: C
Factor: find two numbers that multiply to 0 and add to 3. They are 0 and 3, so the equation is (x + 0)(x + 3) = 0.
The solutions are x = 0 and x = −3.
Their sum is −3. (Shortcut: for x² + bx + c = 0 the sum of the solutions is −b.)
- Question 7Medium
x² + 8x + 12 = 0
What is the sum of the solutions to the equation above?
Show answer and explanation
Answer: A
Factor: find two numbers that multiply to 12 and add to 8. They are 2 and 6, so the equation is (x + 2)(x + 6) = 0.
The solutions are x = −2 and x = −6.
Their sum is −8. (Shortcut: for x² + bx + c = 0 the sum of the solutions is −b.)
- Question 8Medium
x² + x − 6 = 0
What is the sum of the solutions to the equation above?
Show answer and explanation
Answer: A
Factor: find two numbers that multiply to −6 and add to 1. They are −2 and 3, so the equation is (x − 2)(x + 3) = 0.
The solutions are x = 2 and x = −3.
Their sum is −1. (Shortcut: for x² + bx + c = 0 the sum of the solutions is −b.)
- Question 9Medium
x² + 5x − 6 = 0
What is the sum of the solutions to the equation above?
Show answer and explanation
Answer: A
Factor: find two numbers that multiply to −6 and add to 5. They are 6 and −1, so the equation is (x + 6)(x − 1) = 0.
The solutions are x = −6 and x = 1.
Their sum is −5. (Shortcut: for x² + bx + c = 0 the sum of the solutions is −b.)
- Question 10Medium
x² − x + 0 = 0
What is the positive solution to the equation above?
Show answer and explanation
Answer: B
Factor: find two numbers that multiply to 0 and add to −1. They are 0 and −1, so the equation is (x + 0)(x − 1) = 0.
The solutions are x = 0 and x = 1.
The positive solution is 1.
More Advanced Math practice
Questions people ask
Is this quadratic equations practice like the real SAT?
The questions follow the format of the digital SAT's Advanced Math domain: four answer choices built from common mistakes. They are written for this site; they are not official College Board questions.
How many practice questions are there?
As many as you want. Each math question is generated from a template with new numbers, and its answer is checked by code. Use "New set" for another ten.
Where is the domain on the test?
Advanced Math is one of the four content domains of the SAT Math section, alongside Algebra, Problem-Solving and Data Analysis, Geometry and Trigonometry.