The expression x^2 - 9 can be factored as (x - 3)(x + 3).
Correct Answer
True
This is a difference of squares, which factors into the product of the sum and difference of the square roots of the terms.
Question 2
A line with an undefined slope is a horizontal line.
Correct Answer
False
A line with an undefined slope is a vertical line. A horizontal line has a slope of zero.
Question 3
The graph of y = |x - 2| is the graph of y = |x| shifted 2 units up.
Correct Answer
False
The graph of y = |x - 2| is the graph of y = |x| shifted 2 units to the right. A shift up would be represented by y = |x| + 2.
Question 4
The expression x^-2 is equivalent to -x^2.
Correct Answer
False
A negative exponent indicates the reciprocal of the base raised to the positive exponent, so x^-2 is equivalent to 1/x^2, not -x^2.
Question 5
The quadratic formula can only be used to solve quadratic equations that are not factorable.
Correct Answer
False
The quadratic formula can be used to solve any quadratic equation, regardless of whether it is factorable or not.
Question 6
The expression (x^2)^3 simplifies to x^5.
Correct Answer
False
When raising a power to another power, the exponents are multiplied, so (x^2)^3 simplifies to x^(2*3) = x^6.
Question 7
The vertex of the parabola y = (x - 3)^2 + 2 is (3, 2).
Correct Answer
True
A quadratic function in vertex form y = a(x - h)^2 + k has its vertex at the point (h, k). In this case, h = 3 and k = 2.
Question 8
The equation 3(x - 2) = 3x - 6 has infinitely many solutions.
Correct Answer
True
Distributing the 3 on the left side gives 3x - 6 = 3x - 6, which is an identity, meaning any real number for x will satisfy the equation.
Question 9
The range of the function f(x) = x^2 is all real numbers.
Correct Answer
False
For f(x) = x^2, the output values (y-values) are always non-negative, meaning the range is all real numbers greater than or equal to zero, not all real numbers.
Question 10
If the discriminant of a quadratic equation is negative, the equation has two distinct real solutions.
Correct Answer
False
If the discriminant (b^2 - 4ac) of a quadratic equation is negative, the equation has two distinct complex solutions, not real solutions.
Question 11
For any non-zero real number a, a^0 = 1.
Correct Answer
True
By definition, any non-zero number raised to the power of zero is equal to 1.
Question 12
The domain of the function f(x) = sqrt(x - 4) includes all real numbers greater than or equal to 4.
Correct Answer
True
For the square root of a real number to be defined, the expression under the radical must be non-negative. So, x - 4 >= 0, which means x >= 4.
Question 13
The expression sqrt(18) simplifies to 3*sqrt(2).
Correct Answer
True
The number 18 can be factored as 9 * 2. Since 9 is a perfect square, sqrt(18) = sqrt(9 * 2) = sqrt(9) * sqrt(2) = 3 * sqrt(2).
Question 14
The expression x^2 + 4 is factorable over the real numbers.
Correct Answer
False
The sum of two squares, such as x^2 + 4, is not factorable into linear factors with real coefficients. It is considered prime over the real numbers.
Question 15
A system of two linear equations with no solution will have graphs that are parallel lines.
Correct Answer
True
Parallel lines have the same slope but different y-intercepts, meaning they never intersect, indicating no common solution.
Question 16
The solution to the system y = 2x + 1 and y = -x + 4 is (1, 3).
Correct Answer
True
Substituting x=1 into the first equation gives y = 2(1) + 1 = 3. Substituting x=1 into the second equation gives y = -1 + 4 = 3. Since both equations yield y=3 for x=1, the point (1,3) is the solution.
Question 17
The trinomial x^2 + 5x + 6 factors into (x + 2)(x + 3).
Correct Answer
True
To factor x^2 + bx + c, we look for two numbers that multiply to c (6) and add to b (5). These numbers are 2 and 3.
Question 18
The expression 2x^2 + 4x can be factored as 2x(x + 2).
Correct Answer
True
The greatest common factor (GCF) of 2x^2 and 4x is 2x. Factoring out 2x yields 2x(x + 2).
Question 19
A quadratic equation can have exactly one real solution.
Correct Answer
True
A quadratic equation has exactly one real solution when its discriminant is equal to zero (e.g., a perfect square trinomial like (x-3)^2 = 0).
Question 20
If two lines have different slopes, their system of equations will always have exactly one solution.
Correct Answer
True
Lines with different slopes will always intersect at exactly one point, representing a unique solution to the system.
Question 21
Every vertical line intersects the graph of a function at most once.
Correct Answer
True
This is the definition of the vertical line test, which determines if a relation is a function. If a vertical line intersects the graph more than once, it means an x-value corresponds to multiple y-values, violating the definition of a function.
Question 22
The sum sqrt(3) + sqrt(12) simplifies to 3*sqrt(3).
Correct Answer
True
First simplify sqrt(12) as sqrt(4 * 3) = 2*sqrt(3). Then, sqrt(3) + 2*sqrt(3) = 3*sqrt(3).