Advanced Math practice questions
Quadratics, polynomials, exponentials, and rational expressions. Factoring is the fastest route through most of these, so practise recognising the form before reaching for the quadratic formula.
Advanced Math questions
- 1Quadratic equationsStandard
Which value of x is a solution to x^2 - 5x + 6 = 0?
- A1
- B2
- C4
- D6
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Answer: 2
Factor the expression: (x - 2)(x - 3) = 0. The solutions are 2 and 3, so 2 is the listed solution.
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Which pair gives all solutions to (x + 4)(x - 2) = 0?
- Ax = -4 or x = 2
- Bx = 4 or x = -2
- Cx = -8 or x = 0
- Dx = 6 only
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Answer: x = -4 or x = 2
A product is zero when either factor is zero: x + 4 = 0 or x - 2 = 0.
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What is the vertex of y = 2(x - 3)^2 + 1?
- A(3, -1)
- B(3, 1)
- C(-3, 1)
- D(1, 3)
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Answer: (3, 1)
In vertex form a(x - h)^2 + k, the vertex is (h, k). Here h = 3 and k = 1.
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Enter the value of x: 3^x = 27.
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Answer: 3
Because 27 = 3^3, x must equal 3.
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For x not equal to 3, which expression is equivalent to (x^2 - 9) / (x - 3)?
- A1
- Bx + 3
- Cx - 3
- Dx^2 + 3
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Answer: x + 3
Factor the numerator as (x - 3)(x + 3), then cancel the nonzero factor x - 3.
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If h(x) = (x + 2)^2 - 5, what is h(1)?
- A1
- B4
- C-4
- D9
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Answer: 4
h(1) = (1 + 2)^2 - 5 = 9 - 5 = 4.
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Enter the positive solution to x^2 = 49.
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Answer: 7
Both 7 and -7 square to 49, but the question asks for the positive solution: 7.
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Which expression factors x^2 + 6x + 5?
- A(x + 1)(x + 5)
- B(x - 2)(x - 3)
- C(x + 2)(x + 3)
- D(x - 1)(x - 5)
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Answer: (x + 1)(x + 5)
The factors must multiply to 5 and add to 6. Those numbers are 1 and 5.
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What is the value of x if 2^(x + 1) = 16?
- A2
- B3
- C5
- D4
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Answer: 3
Since 16 = 2^4, x + 1 = 4. Therefore x = 3.
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The roots of (x - 1)(x + 5) = 0 are 1 and -5. What is their sum?
- A-6
- B-4
- C4
- D6
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Answer: -4
Add the two roots: 1 + (-5) = -4.
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Which expression is a factor of x^3 - 8?
- Ax - 2
- Bx + 2
- Cx - 8
- Dx + 8
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Answer: x - 2
Substituting x = 2 gives 2^3 - 8 = 0, so x - 2 is a factor.
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How many real solutions does x^2 + 4x + 5 = 0 have?
- AInfinitely many
- B2
- C0
- D1
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Answer: 0
The discriminant is 4^2 - 4(1)(5) = 16 - 20 = -4. A negative discriminant means there are no real solutions.
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What is the minimum value of y = x^2 - 6x + 11?
- A3
- B11
- C-2
- D2
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Answer: 2
Complete the square: y = (x - 3)^2 + 2. The squared term is never negative, so the smallest value of y is 2, reached at x = 3.
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Which expression factors x^2 - 7x + 12?
- A(x - 3)(x - 4)
- B(x - 2)(x - 6)
- C(x - 1)(x - 12)
- D(x + 3)(x + 4)
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Answer: (x - 3)(x - 4)
Two numbers multiplying to 12 and adding to -7 are -3 and -4, so the factors are (x - 3)(x - 4).
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Which expression is equivalent to (2x^3)(5x^4)?
- A7x^12
- B10x^7
- C7x^7
- D10x^12
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Answer: 10x^7
Multiply the coefficients: 2 x 5 = 10. Add the exponents on the shared base: x^3 x^4 = x^7.
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Enter the value of 16^(1/2).
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Answer: 4
An exponent of 1/2 means the positive square root, and the square root of 16 is 4.
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For x not equal to -2, which expression is equivalent to (x^2 + 5x + 6) / (x + 2)?
- Ax + 3
- Bx^2 + 3
- Cx - 3
- Dx + 2
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Answer: x + 3
Factor the numerator as (x + 2)(x + 3), then cancel the nonzero factor x + 2.
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The solutions to x^2 + bx + 12 = 0 are 2 and 6. What is the value of b?
- A-8
- B8
- C-12
- D12
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Answer: -8
If the solutions are 2 and 6, the equation factors as (x - 2)(x - 6) = x^2 - 8x + 12. Comparing with x^2 + bx + 12 gives b = -8.
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A population of 300 bacteria doubles every hour. Enter the population after 3 hours.
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Answer: 2400
Doubling three times multiplies by 2^3 = 8, and 300 x 8 = 2400.
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If g(x) = x^2 - 4x, what is the value of g(-2)?
- A-4
- B12
- C4
- D-12
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Answer: 12
Substitute x = -2: (-2)^2 - 4(-2) = 4 + 8 = 12.
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Which expression is equivalent to (3x - 5)(3x + 5)?
- A3x^2 - 25
- B9x^2 + 25
- C9x^2 - 30x - 25
- D9x^2 - 25
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Answer: 9x^2 - 25
This is a difference of squares: (a - b)(a + b) = a^2 - b^2, with a = 3x and b = 5, giving 9x^2 - 25.
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At how many points do the graphs of y = x^2 and y = 2x - 1 intersect?
- A1
- B0
- CInfinitely many
- D2
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Answer: 1
Set x^2 = 2x - 1, which rearranges to x^2 - 2x + 1 = 0, or (x - 1)^2 = 0. The single repeated solution x = 1 means the line touches the parabola at exactly one point.
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Which expression gives the solutions to 2x^2 + 3x - 4 = 0?
- A(3 +/- sqrt(41)) / 4
- B(-3 +/- sqrt(41)) / 4
- C(-3 +/- sqrt(41)) / 2
- D(-3 +/- sqrt(23)) / 4
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Answer: (-3 +/- sqrt(41)) / 4
This quadratic does not factor, so apply the quadratic formula, x = (-b +/- sqrt(b^2 - 4ac)) / (2a), with a = 2, b = 3, c = -4. The discriminant is 9 - 4(2)(-4) = 9 + 32 = 41, and 2a = 4.
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What is the axis of symmetry of the parabola y = x^2 - 10x + 7?
- Ax = -5
- Bx = 7
- Cx = 5
- Dx = 10
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Answer: x = 5
The axis of symmetry of y = ax^2 + bx + c is x = -b / (2a). Here that is -(-10) / (2 x 1) = 5.
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What is the value of 5^(-2)?
- A-25
- B1/25
- C-10
- D25
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Answer: 1/25
A negative exponent means the reciprocal of the positive power: 5^(-2) = 1 / 5^2 = 1/25. It does not make the result negative.
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If x is not zero, which expression is equivalent to (3x^4) / (3x^4)?
- A1
- B0
- Cx
- D3
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Answer: 1
Any nonzero quantity divided by itself is 1. Using the quotient rule gives x^(4-4) = x^0, and x^0 = 1 for every nonzero x.
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The graph of y = (x - 2)(x + 6) crosses the x-axis at two points. What is the x-coordinate of its vertex?
- A-6
- B2
- C-2
- D4
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Answer: -2
The roots are x = 2 and x = -6, and a parabola is symmetric about the midpoint of its roots: (2 + (-6)) / 2 = -2.
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A sample of 400 grams decays so that 20 percent is lost each year. Which expression gives the mass remaining after t years?
- A400(1.20)^t
- B400 - 0.20t
- C400(0.20)^t
- D400(0.80)^t
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Answer: 400(0.80)^t
Losing 20 percent leaves 80 percent, so each year multiplies the mass by 0.80. Repeated multiplication over t years gives 400(0.80)^t.
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For which value of x is the expression (x + 1) / (x^2 - 4x + 3) undefined at x = 3 but defined at x = 2?
- AThe expression is undefined at x = 2 and defined at x = 3.
- BThe expression is undefined only at x = -1.
- CThe expression is undefined at x = 3 and x = 1, and defined at x = 2.
- DThe expression is defined for every real value of x.
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Answer: The expression is undefined at x = 3 and x = 1, and defined at x = 2.
A rational expression is undefined where its denominator is zero. Factoring gives x^2 - 4x + 3 = (x - 3)(x - 1), so the excluded values are 3 and 1. At x = 2 the denominator is -1, which is fine.
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Which expression is equivalent to sqrt(50)?
- A2 sqrt(5)
- B25 sqrt(2)
- C5 sqrt(2)
- D10 sqrt(5)
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Answer: 5 sqrt(2)
Factor out the largest perfect square: 50 = 25 x 2, so sqrt(50) = sqrt(25) x sqrt(2) = 5 sqrt(2).
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If p(x) = x^3 - 2x, enter the value of p(3).
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Answer: 21
Substitute x = 3: 3^3 - 2(3) = 27 - 6 = 21.
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A polynomial q(x) satisfies q(4) = 0. Which statement must be true?
- Aq(x) has exactly one real root
- Bq(0) = 4
- Cx + 4 is a factor of q(x)
- Dx - 4 is a factor of q(x)
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Answer: x - 4 is a factor of q(x)
By the factor theorem, q(a) = 0 exactly when x - a divides q(x). Here a = 4, so x - 4 is a factor. Nothing follows about the number of roots or about q(0).
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What is the sum of the solutions to x^2 - 9x + 14 = 0?
- A14
- B5
- C9
- D-9
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Answer: 9
For ax^2 + bx + c = 0 the solutions sum to -b/a, which is 9 here. Factoring confirms it: (x - 2)(x - 7) gives 2 and 7.
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If 4^x = 8, what is the value of x?
- A1.5
- B3
- C0.5
- D2
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Answer: 1.5
Write both sides on base 2: 4^x = 2^(2x) and 8 = 2^3. So 2x = 3 and x = 1.5. Matching bases is the move that makes this a one-line problem.
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For which value of x is (x^2 - 1) / (x^2 - x - 2) undefined?
- Ax = 2 and x = -1
- BIt is defined for all x.
- Cx = 2 only
- Dx = 1 and x = -1
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Answer: x = 2 and x = -1
A quotient is undefined where the denominator is zero. Factoring gives (x - 2)(x + 1), so x = 2 and x = -1. The numerator's zeros are irrelevant to whether the expression is defined.
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The parabola y = x^2 + bx + 7 has its vertex at x = 3. What is b?
- A-6
- B6
- C-3
- D3
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Answer: -6
The vertex sits at x = -b / (2a) with a = 1, so -b / 2 = 3 and b = -6.
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Enter the value of x if sqrt(x + 7) = 5.
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Answer: 18
Square both sides: x + 7 = 25, so x = 18. Checking in the original equation confirms it, which matters because squaring can introduce solutions that do not satisfy the original.
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A culture of 5,000 cells grows by 20 percent per hour. Which expression gives the population after h hours?
- A5000 + 0.20h
- B5000(1.20)^h
- C5000(20)^h
- D5000(0.20)^h
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Answer: 5000(1.20)^h
Growing by 20 percent multiplies by 1.20 each hour, and repeating that h times gives 5000(1.20)^h. The additive form would describe a constant increase of 0.2 cells, not 20 percent.
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Which expression is equivalent to (2x + 3)^2?
- A2x^2 + 12x + 9
- B4x^2 + 6x + 9
- C4x^2 + 9
- D4x^2 + 12x + 9
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Answer: 4x^2 + 12x + 9
Squaring a binomial gives a^2 + 2ab + b^2, so (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9. Dropping the middle term is the most common error.
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If f(x) = x^2 - 4 and f(a) = 21, what are the possible values of a?
- A17 and -17
- B25 and -25
- C5 only
- D5 and -5
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Answer: 5 and -5
Solving x^2 - 4 = 21 gives x^2 = 25, so a is 5 or -5. Reporting only the positive root answers half the question.
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Which expression factors 2x^2 + 7x + 3?
- A(x + 1)(x + 3)
- B(2x + 3)(x + 1)
- C(2x + 1)(x + 3)
- D(2x + 7)(x + 3)
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Answer: (2x + 1)(x + 3)
Expanding (2x + 1)(x + 3) gives 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3. The alternative pairing gives a middle term of 5x, not 7x.
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For what value of c does x^2 - 6x + c = 0 have exactly one real solution?
- A6
- B-9
- C9
- D3
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Answer: 9
One real solution means the discriminant is zero: 36 - 4c = 0, so c = 9. The equation is then (x - 3)^2 = 0.
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Which expression is equivalent to (8x^6)^(1/3)?
- A2x^3
- B8x^2
- C2x^2
- D4x^2
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Answer: 2x^2
A cube root applies to both factors: the cube root of 8 is 2, and x^6 raised to 1/3 gives x^2 because the exponents multiply.
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For x not equal to 4, which expression is equivalent to (x^2 - 16) / (4 - x)?
- Ax + 4
- B-(x - 4)
- Cx - 4
- D-(x + 4)
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Answer: -(x + 4)
Factor the numerator as (x - 4)(x + 4). Since 4 - x = -(x - 4), the quotient is -(x + 4). Missing the sign flip in the denominator is the trap.
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A quantity halves every 6 hours. What fraction of the original remains after 24 hours?
- A1/4
- B1/8
- C1/24
- D1/16
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Answer: 1/16
Twenty-four hours is four half-lives, so the quantity is multiplied by (1/2)^4 = 1/16.
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If h(x) = 2x^2 - x, enter the value of h(-3).
Student-produced response: write your own answer rather than choosing one.
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Answer: 21
Substitute x = -3: 2(9) - (-3) = 18 + 3 = 21. The minus sign in front of x makes the second term positive here.
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The graph of y = (x + 2)(x - 1)(x - 4) crosses the x-axis at how many points?
- A1
- B4
- C2
- D3
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Answer: 3
Each distinct factor gives one x-intercept: x = -2, x = 1 and x = 4. The three factors have no repeats, so all three are crossings.
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The parabola y = -(x - 5)^2 + 12 has which of the following?
- AA minimum of -12 at x = 5
- BA maximum of 5 at x = 12
- CA minimum of 12 at x = 5
- DA maximum of 12 at x = 5
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Answer: A maximum of 12 at x = 5
The negative sign in front of the squared term opens the parabola downward, so the vertex is a maximum. Vertex form gives the vertex at (5, 12).
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What is the product of the solutions to x^2 - 4x - 21 = 0?
- A-4
- B4
- C21
- D-21
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Answer: -21
For ax^2 + bx + c = 0 the solutions multiply to c/a, which is -21. Factoring confirms it: (x - 7)(x + 3) gives 7 and -3.
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Which expression is equivalent to (x^5 y^2) / (x^2 y^5), for x and y not zero?
- Ax^7 / y^7
- B1 / (x^3 y^3)
- Cx^3 y^3
- Dx^3 / y^3
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Answer: x^3 / y^3
Subtract exponents on each base: x^(5-2) = x^3 and y^(2-5) = y^-3, which is 1/y^3. Together that is x^3 / y^3.
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The graph of y = x^2 + kx + 16 touches the x-axis at exactly one point. What are the possible values of k?
- A8 and -8
- B16 and -16
- C4 and -4
- D8 only
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Answer: 8 and -8
One touching point means the discriminant is zero: k^2 - 64 = 0, so k = 8 or k = -8. Reporting only the positive value answers half the question.
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For x not equal to -3, which expression is equivalent to (2x^2 + 6x) / (x + 3)?
- A2x^2
- B2x
- C2
- D2x + 6
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Answer: 2x
Factor the numerator as 2x(x + 3), then cancel the nonzero factor x + 3, leaving 2x.
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An investment of $2,000 grows by 5 percent each year. Which expression gives its value after t years?
- A2000(5)^t
- B2000(0.05)^t
- C2000(1.05)^t
- D2000 + 0.05t
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Answer: 2000(1.05)^t
Growing by 5 percent multiplies by 1.05 each year, repeated t times. The additive form would add five cents a year.
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Enter the value of x if sqrt(2x + 1) = x - 1.
Student-produced response: write your own answer rather than choosing one.
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Answer: 4
Square both sides: 2x + 1 = x^2 - 2x + 1, so x^2 - 4x = 0 and x is 0 or 4. Checking in the original, x = 0 gives 1 = -1, which is false, so only x = 4 works. Squaring can create solutions that do not satisfy the original.
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Which expression is a factor of x^2 - 5x - 24?
- Ax + 4
- Bx - 6
- Cx - 8
- Dx + 8
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Answer: x - 8
Two numbers multiplying to -24 and adding to -5 are -8 and 3, so the factors are (x - 8)(x + 3).
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If f(x) = 2x + 1 and g(x) = x^2, what is g(f(2))?
- A9
- B17
- C25
- D5
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Answer: 25
Work from the inside: f(2) = 5, then g(5) = 25. Applying g first would give f(4) = 9, which is the trap.
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