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SAT · Math · Advanced Math
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.
Question 1
A scaffolding's cross-brace lengths satisfy 3x² − 15x + 12 = 0. What is the product of the solutions of this equation?
-4
4
5
12
Answer: B. 4
For ax² + bx + c = 0, the product of the solutions is c/a = 12/3 = 4 (B). Choice A has the wrong sign; C is the sum of the solutions, −b/a = 5; D is the constant c without dividing by a.
Question 2
2x² + 7x = 15 A leisure centre's check-in model leads to the equation above. What is the positive solution of the equation?
−5
−3/2
3/2
3
Answer: C. 3/2
Rewrite as 2x² + 7x − 15 = 0 = (2x − 3)(x + 5), so x = 3/2 or x = −5, and the positive solution is 3/2 (C). A is the negative solution; B has the sign of the factor 2x − 3 reversed; D solves x − 3 = 0, ignoring the leading coefficient 2.
Question 3
The number of ticket sections x in an arena's lower bowl satisfies x² − 4x = 21. What is the positive solution of the equation?
−7
3
4
7
Answer: D. 7
Rewrite as x² − 4x − 21 = 0, which factors as (x − 7)(x + 3) = 0. The solutions are 7 and −3, so the positive one is 7. A reads the factor x − 7 as the solution −7; B is the absolute value of the negative solution; C is the coefficient 4, which is not a solution (16 − 16 ≠ 21).
Question 4
What is the positive solution to the equation x² - 7x - 18 = 0?
Enter your own answer.
Answer: 9
Factor: x² - 7x - 18 = (x - 9)(x + 2) = 0, since 9 × (-2) = -18 and 9 + (-2) = 7. The solutions are 9 and -2, so the positive solution is 9.
Question 5
How many distinct real solutions does the equation -x² - 6x - 9 = 0 have?
Exactly two
Zero
Infinitely many
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 few questions show the style. The free scan shows which SAT skills cost you the most points, then builds a plan around them.