Question 1
What is the positive solution to the equation x² + 5x - 36 = 0?
- A.
-9
- B.
-5
- C.
-4
- D.
4
Answer: D. 4
Factor: (x - 4)(x + 9) = 0, so x = 4 or x = -9. The positive solution is 4.
SAT · Math · Advanced Math
Nonlinear equations in one variable and systems of equations in two variables
Nonlinear equations in one variable and systems of equations in two variables is part of Advanced Math in the SAT Math section. The questions below are written in the style of the real test, each with the answer and a worked explanation.
Question types you will see here include solve a quadratic by factoring or formula, number of real solutions via the discriminant, linear and quadratic system and rational equation and extraneous solutions.
Common traps: sign error on a factor (x - 3 gives -3); giving both roots when one is asked for; forgetting to move every term to one side before factoring.
Sample questions
Question 2
In the xy-plane, the line y = 2x + k intersects the parabola y = x² - 6x + 10 at exactly one point. What is the value of k?
- A.
-6
- B.
6
- C.
10
- D.
24
Answer: A. -6
Set x² - 6x + 10 = 2x + k, so x² - 8x + (10 - k) = 0. One intersection means the discriminant is 0: 64 - 4(10 - k) = 24 + 4k = 0, so k = -6, and A is correct. B makes a sign error, writing the discriminant as 24 - 4k; C is the parabola's y-intercept, which the tangent line does not need to share; D stops at the constant term of the discriminant, 24 + 4k, without solving for k.
- A.
Question 3
A ceremonial cannon at a fort fires a ball that can land at positions x, in hundreds of metres, satisfying |2x − 10| = 6. What is the distance between the two solutions?
- A.
2
- B.
3
- C.
6
- D.
8
Answer: C. 6
Solving 2x − 10 = 6 gives x = 8 and 2x − 10 = −6 gives x = 2, so the solutions are 6 apart (C). A and D are the individual solutions 2 and 8; B is the deviation 3 of each solution from the centre 5.
- A.
Question 4
√(2x + 6) + 1 = x A web host's overage model leads to the equation above. What is the solution set of the equation?
- A.
{−1}
- B.
{5}
- C.
{−1, 5}
- D.
There is no solution.
Answer: B. {5}
Isolate the root and square: 2x + 6 = (x − 1)², so x² − 4x − 5 = 0 and x = 5 or x = −1. Check: x = 5 gives √16 + 1 = 5, true; x = −1 gives √4 + 1 = 3, not −1. So the set is {5} (B). A keeps only the extraneous root; C skips the check; D rejects a valid root.
- A.
Question 5
How many distinct real solutions does the equation -x² - 6x - 9 = 0 have?
- A.
Exactly two
- B.
Zero
- C.
Infinitely many
- D.
Exactly one
Answer: D. Exactly one
The discriminant is b² - 4ac = (-6)² - 4(-1)(-9) = 0, which is zero, so there is exactly one real solution.
- A.
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