Question ID b8caaf84
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Equivalent
expressions
ID: b8caaf84
If and , which of the following is
equivalent to ?
A.
B.
C.
D.
ID: b8caaf84 Answer
Correct Answer: B
Rationale
Choice B is correct. It’s given that and . Substituting the values for p and v into the expression
yields . Multiplying the terms yields .
Using the distributive property to rewrite yields . Therefore, the entire expression can be represented as
. Combining like terms yields .
Choice A is incorrect and may result from subtracting, instead of adding, the term . Choice C is incorrect. This is the
result of multiplying the terms . Choice D is incorrect and may result from distributing 2, instead of , to the
term .
Question Difficulty: Medium
Question ID 371cbf6b
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Equivalent
expressions
ID: 371cbf6b
The equation above is true for all x, where a and b are
constants. What is the value of ab ?
A. 18
B. 20
C. 24
D. 40
ID: 371cbf6b Answer
Correct Answer: C
Rationale
Choice C is correct. If the equation is true for all x, then the expressions on both sides of the equation will be equivalent.
Multiplying the polynomials on the left-hand side of the equation gives . On the right-
hand side of the equation, the only -term is . Since the expressions on both sides of the equation are equivalent, it
follows that , which can be rewritten as . Therefore, , which
gives .
Choice A is incorrect. If , then the coefficient of on the left-hand side of the equation would be ,
which doesn’t equal the coefficient of , , on the right-hand side. Choice B is incorrect. If , then the coefficient
of on the left-hand side of the equation would be , which doesn’t equal the coefficient of , , on the
right-hand side. Choice D is incorrect. If , then the coefficient of on the left-hand side of the equation would be
, which doesn’t equal the coefficient of , , on the right-hand side.
Question Difficulty: Hard
Question ID a05bd3a4
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Equivalent
expressions
ID: a05bd3a4
Which of the following expressions is
equivalent to ?
A.
B.
C.
D.
ID: a05bd3a4 Answer
Correct Answer: C
Rationale
Choice C is correct. The expression can be written as a difference of squares x2 – y2, which can be factored as (x + y)(x – y).
Here, y2 = 5, so , and the expression therefore factors as .
Choices A and B are incorrect and may result from misunderstanding how to factor a difference of squares. Choice D is
incorrect; (x + 5)(x – 1) can be rewritten as x2 + 4x – 5, which is not equivalent to the original expression.
Question Difficulty: Medium
Question ID a5663025
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Nonlinear equations
in one variable and
systems of
equations in two
variables
ID: a5663025
A system of equations consists of a quadratic equation and a linear equation. The
equations in this system are graphed in the xy-plane above. How many solutions
does this system have?
A. 0
B. 1
C. 2
D. 3
ID: a5663025 Answer
Correct Answer: C
Rationale
Choice C is correct. The solutions to a system of two equations correspond to points where the graphs of the equations
intersect. The given graphs intersect at 2 points; therefore, the system has 2 solutions.
Choice A is incorrect because the graphs intersect. Choice B is incorrect because the graphs intersect more than once.
Choice D is incorrect. It’s not possible for the graph of a quadratic equation and the graph of a linear equation to intersect
more than twice.
Question Difficulty: Medium
Question ID d0a7871e
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Nonlinear equations
in one variable and
systems of
equations in two
variables
ID: d0a7871e
If is a solution to the system of equations above, which of the
following could be the value of x ?
A. –1
B. 0
C. 2
D. 3
ID: d0a7871e Answer
Correct Answer: A
Rationale
Choice A is correct. It is given that y = x + 1 and y = x2 + x. Setting the values for y equal to each other yields x + 1 = x2 + x.
Subtracting x from each side of this equation yields x2 = 1. Therefore, x can equal 1 or –1. Of these, only –1 is given as a
choice.
Choice B is incorrect. If x = 0, then x + 1 = 1, but x2 + x = 02 + 0 = 0 ≠︀1. Choice C is incorrect. If x = 2, then x + 1 = 3, but x2 + x
= 22 + 2 = 6 ≠︀3. Choice D is incorrect. If x = 3, then x + 1 = 4, but x2 + x = 32 + 3 = 12 ≠︀4.
Question Difficulty: Medium
Question ID 3a9d60b2
Assessment Test Domain Skill Difficulty
SAT Math Advanced Math Nonlinear equations
in one variable and
systems of
equations in two
variables
ID: 3a9d60b2
What is the positive solution to the given equation?
ID: 3a9d60b2 Answer
Correct Answer: 9
Rationale
The correct answer is . The given equation can be rewritten as . Dividing each side of this equation by
yields . By the definition of absolute value, if , then or . Subtracting from
each side of the equation yields . Dividing each side of this equation by yields . Similarly,
subtracting from each side of the equation yields . Dividing each side of this equation by
yields . Therefore, since the two solutions to the given equation are and , the positive solution to the given
equation is .
Question Difficulty: Hard
