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
- 57Exponential growth and decayStandard
A culture starts with 400 cells and doubles every 6 hours. How many cells are there after 24 hours?
- A6,400
- B1,600
- C3,200
- D9,600
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Answer: 6,400
24 hours is four 6-hour periods, so the count doubles four times: 400 x 2^4 = 400 x 16 = 6,400. Multiplying by 4 rather than doubling four times gives 1,600, and counting three doublings gives 3,200.
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What is the positive difference between the solutions to x^2 - 10x + 21 = 0?
- A10
- B4
- C21
- D3
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Answer: 4
The expression factors as (x - 3)(x - 7), so the solutions are 3 and 7 and the difference is 4. The other choices are the sum of the solutions, their product, and one solution on its own.
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For x not equal to -4 or -3, which expression is equivalent to (x^2 - 9) / (x^2 + 7x + 12)?
- A(x - 3) / (x - 4)
- B(x + 3) / (x + 4)
- C(x - 3) / (x + 4)
- D(x - 9) / (x + 12)
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Answer: (x - 3) / (x + 4)
The numerator is a difference of squares, (x - 3)(x + 3), and the denominator factors as (x + 3)(x + 4). The common factor x + 3 cancels, leaving (x - 3) / (x + 4). The last choice cancels the x terms instead, which is not something cancellation permits.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 60Exponent rulesStandard
Which expression is equivalent to (2x^3 y)^4 / (4x^5 y^2), where x and y are positive?
- A8x^7 y^2
- B4x^12 y^2
- C2x^7 y^2
- D4x^7 y^2
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Answer: 4x^7 y^2
The power of 4 applies to every factor: (2x^3 y)^4 = 16x^12 y^4. Dividing by 4x^5 y^2 gives 16/4 = 4, x^(12-5) = x^7 and y^(4-2) = y^2. Leaving the exponent on x undivided gives x^12; forgetting to raise the 2 gives the wrong coefficient.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 61Quadratic functionsChallenge
What is the minimum value of f(x) = x^2 - 6x + 11?
- A2
- B3
- C11
- D-2
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Answer: 2
Completing the square gives f(x) = (x - 3)^2 + 2. A square is never negative, so the smallest value is 2, reached at x = 3. The choice 3 is where the minimum occurs rather than what it is, and 11 is the value at x = 0.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 62Roots of polynomialsChallenge
The polynomial p(x) = x^3 - 4x^2 - 11x + 30 has 2 among its roots. What are the other two?
- A5 and 3
- B5 and -3
- C-5 and 3
- D-5 and -3
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Answer: 5 and -3
Dividing p(x) by (x - 2) leaves x^2 - 2x - 15, which factors as (x - 5)(x + 3). So the other roots are 5 and -3. Checking is quick: p(5) = 125 - 100 - 55 + 30 = 0 and p(-3) = -27 - 36 + 33 + 30 = 0.
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What is the solution to the square root of (3x + 4) equal to 5?
- A3
- B9
- C7
- D21
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Answer: 7
Squaring both sides gives 3x + 4 = 25, so 3x = 21 and x = 7. Squaring can introduce false solutions, so check: 3(7) + 4 = 25 and the square root of 25 is 5. Squaring only one side gives 21.
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The graphs of y = x^2 - 2x and y = 3x - 4 intersect at two points. What is the sum of the x-coordinates of those points?
- A4
- B-5
- C1
- D5
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Answer: 5
Setting the expressions equal gives x^2 - 2x = 3x - 4, so x^2 - 5x + 4 = 0 and (x - 1)(x - 4) = 0. The x-coordinates are 1 and 4, summing to 5. Reading the sum off the equation as -5 uses the wrong sign.
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What are the solutions to (2x - 6)(x + 4) = 0?
- A3 and -4
- B-3 and 4
- C6 and -4
- D2 and -4
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Answer: 3 and -4
A product is zero when a factor is zero. From 2x - 6 = 0 comes x = 3, and from x + 4 = 0 comes x = -4. Reading 6 straight off the first factor forgets to divide by the 2.
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If x is positive and x^2 = 49/4, what is the value of x?
- A24.5
- B3.5
- C12.25
- D7
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Answer: 3.5
Taking the positive square root of both sides gives x = 7/2 = 3.5. The choice 12.25 is x^2 expressed as a decimal, and 7 takes the root of the numerator while leaving the denominator alone.
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For positive x and y, which expression is equivalent to the cube root of x^6 y^3?
- Ax^3 y
- Bx^2 y^3
- Cx^2 y
- Dx^18 y^9
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Answer: x^2 y
A cube root is the power 1/3, and a power of a power multiplies the exponents: x^(6/3) y^(3/3) = x^2 y. The last choice multiplies the exponents by 3 instead of dividing, which is the inverse operation.
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Which expression is equivalent to (x + 5)(x - 5) + 25?
- Ax^2 + 25
- Bx^2 - 50
- Cx^2 + 10x
- Dx^2
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Answer: x^2
The product is a difference of squares, x^2 - 25, and adding 25 cancels the constant exactly, leaving x^2. Expanding the bracket as if it were (x + 5)^2 produces the choice with a 10x term.
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How many real solutions does the system y = x^2 + 3 and y = 4x have?
- ATwo
- BOne
- CNone
- DInfinitely many
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Answer: Two
Setting the expressions equal gives x^2 - 4x + 3 = 0, which factors as (x - 1)(x - 3), so the curves meet where x is 1 and where x is 3. A parabola and a line meet twice, once or not at all, and here the discriminant 16 - 12 is positive.
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If 2^(x + 1) = 32, what is the value of x?
- A16
- B4
- C5
- D31
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Answer: 4
Writing 32 as 2^5 makes the exponents comparable: x + 1 = 5, so x = 4. The choice 5 is the exponent on the right rather than the value of x, which is the step this question is built around.
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The graph of y = a(x - 2)^2 + k has vertex (2, 3) and passes through (0, 7). What is the value of a?
- A4
- B2
- C1
- D1/4
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Answer: 1
The vertex gives k = 3. Substituting the point: 7 = a(0 - 2)^2 + 3 = 4a + 3, so 4a = 4 and a = 1. Forgetting to square the -2 gives a = 2.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 72Polynomial factorsChallenge
A polynomial p satisfies p(-3) = 0. Which expression must be a factor of p(x)?
- Ax - 3
- Bx + 1/3
- C3x
- Dx + 3
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Answer: x + 3
By the factor theorem, p(c) = 0 means (x - c) is a factor. Here c is -3, so the factor is x - (-3) = x + 3. Dropping the double negative gives x - 3, which is the factor you would get from p(3) = 0.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 73Factoring quadraticsFoundation
Which expression is the factored form of x^2 + x - 12?
- A(x + 4)(x - 3)
- B(x - 4)(x + 3)
- C(x + 6)(x - 2)
- D(x + 12)(x - 1)
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Answer: (x + 4)(x - 3)
The two numbers must multiply to -12 and add to +1: those are 4 and -3. Swapping the signs gives a middle term of -1 instead, which is the commonest way to get this wrong.
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Which expression is equivalent to (3x - 2)^2?
- A9x^2 + 4
- B9x^2 - 12x + 4
- C9x^2 - 6x + 4
- D3x^2 - 12x + 4
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Answer: 9x^2 - 12x + 4
A squared binomial gives three terms: (3x)^2 = 9x^2, twice the product 2(3x)(-2) = -12x, and (-2)^2 = 4. Squaring each term separately drops the middle one entirely.
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If the square root of (x - 3) equals 4, what is the value of x?
- A7
- B13
- C19
- D1
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Answer: 19
Squaring both sides gives x - 3 = 16, so x = 19. Check it: 19 - 3 = 16 and the square root of 16 is 4. Answering 7 adds 4 to 3 rather than 16.
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For positive x, which expression is equivalent to x raised to the power 3/4?
- Athe cube root of x^4
- Bx^3 times x^4
- C3x/4
- Dthe fourth root of x^3
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Answer: the fourth root of x^3
In a fractional exponent the numerator is the power and the denominator is the root, so 3/4 means the fourth root of x cubed. The first choice reverses the two, which gives x to the power 4/3.
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At which values of x does f(x) = (x - 5)(x + 2) equal zero?
- A5 and -2
- B-5 and 2
- C5 and 2
- D-5 and -2
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Answer: 5 and -2
Each factor is set to zero in turn: x - 5 = 0 gives 5, and x + 2 = 0 gives -2. The sign in the bracket is the opposite of the zero, which is the step every wrong choice here skips.
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For which positive value of k does x^2 + kx + 16 = 0 have exactly one real solution?
- A4
- B8
- C16
- D32
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Answer: 8
One real solution means the discriminant is zero: k^2 - 4(1)(16) = k^2 - 64 = 0, so k is 8 or -8, and the positive value is 8. Then the equation is x^2 + 8x + 16 = (x + 4)^2, whose only root is -4.
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If 3^a multiplied by 3^5 equals 3^12, enter the value of a.
Student-produced response: write your own answer rather than choosing one.
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Answer: 7
Multiplying powers of the same base adds the exponents, so a + 5 = 12 and a = 7.
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The function h is defined by h(x) = x^2 - 4x. Enter the value of h(-2).
Student-produced response: write your own answer rather than choosing one.
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Answer: 12
Substituting gives (-2)^2 - 4(-2) = 4 + 8 = 12. Both signs matter: squaring a negative makes it positive, and subtracting a negative adds.
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At which points does the graph of y = x^2 - 8x + 15 cross the x-axis?
- Ax = 3 and x = 5
- Bx = -3 and x = -5
- Cx = 15 and x = 1
- Dx = 8 and x = 15
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Answer: x = 3 and x = 5
The graph crosses where y = 0, and x^2 - 8x + 15 factors as (x - 3)(x - 5). The signs in the brackets are the opposite of the roots, which is what the second choice gets wrong.
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For positive x and y, which expression is equivalent to (x^5 y^2) / (x^2 y^5)?
- Ax^3 y^3
- Bx^3 / y^3
- Cy^3 / x^3
- Dx^7 y^7
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Answer: x^3 / y^3
Dividing subtracts exponents: x^(5-2) = x^3 and y^(2-5) = y^-3, and a negative exponent means the term moves to the denominator. Multiplying the exponents instead of subtracting gives the last choice.
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For x not equal to 2, which expression is equivalent to (x^2 - 4) / (x - 2)?
- Ax - 2
- Bx^2 - 2
- Cx + 2
- D-2
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Answer: x + 2
The numerator is a difference of squares, (x - 2)(x + 2). Cancelling the common factor leaves x + 2. Cancelling the x terms on their own gives -2, which is not something cancellation permits.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 84Exponential growth and decayChallenge
A population of 5,000 grows by 8 percent a year. Which expression gives the population after t years?
- A5000(0.08)^t
- B5000 + 0.08t
- C5000(1.8)^t
- D5000(1.08)^t
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Answer: 5000(1.08)^t
Growing by 8 percent multiplies by 1.08 each year, so after t years the factor is 1.08^t. Using 0.08 would shrink the population by 92 percent a year, and adding 0.08t would make the growth linear rather than compounding.
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What are the solutions to x^2 = 6x?
- A6 only
- B-6 and 6
- C0 and 6
- D0 and -6
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Answer: 0 and 6
Bring everything to one side: x^2 - 6x = 0, so x(x - 6) = 0 and x is 0 or 6. Dividing both sides by x is what produces "6 only" — and it silently divides by zero, losing the other solution.
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If f(x) = x^2 + 1, which expression is equal to f(x + 1) - f(x)?
- A1
- Bx^2 + 2x
- C2x
- D2x + 1
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Answer: 2x + 1
f(x + 1) = (x + 1)^2 + 1 = x^2 + 2x + 2. Subtracting f(x) = x^2 + 1 leaves 2x + 1. Treating f(x + 1) as f(x) + 1 gives 1, which is the error the question exists to catch.
How did you do?Saved in this browser. Weak Skills will replay this skill. - 87Factoring quadraticsStandard
The expression x^2 - 13x + 40 factors as (x - 5)(x - k). Enter the value of k.
Student-produced response: write your own answer rather than choosing one.
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Answer: 8
The two constants must multiply to 40 and add to 13, so with one of them 5 the other is 8. Checking: (x - 5)(x - 8) = x^2 - 13x + 40.
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Enter the value of the square root of 144 added to the square root of 25.
Student-produced response: write your own answer rather than choosing one.
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Answer: 17
The two roots are 12 and 5, so the sum is 17. Note that this is not the square root of 169: roots cannot be added under one radical.
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Which expression is equivalent to (x + 7)(x - 2)?
- Ax^2 - 5x - 14
- Bx^2 + 5x - 14
- Cx^2 + 5x + 14
- Dx^2 - 14
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Answer: x^2 + 5x - 14
Multiplying out: x^2 - 2x + 7x - 14 = x^2 + 5x - 14. The middle term comes from adding -2 and 7, and forgetting it entirely gives x^2 - 14.
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What is the value of 8 raised to the power 2/3?
- A16
- B5.33
- C4
- D12
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Answer: 4
The denominator is the root and the numerator the power: the cube root of 8 is 2, and 2 squared is 4. Multiplying 8 by 2/3 gives 5.33, which treats the exponent as a factor.
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What are the solutions to x^2 + 2x - 15 = 0?
- A5 and -3
- B5 and 3
- C-5 and -3
- D-5 and 3
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Answer: -5 and 3
The expression factors as (x + 5)(x - 3), so the solutions are -5 and 3. The signs of the solutions are the opposite of the signs in the brackets, which is what "5 and -3" gets wrong.
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A radioactive sample of 480 mg has a half-life of 4 days. How much remains after 12 days?
- A60 mg
- B160 mg
- C240 mg
- D120 mg
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Answer: 60 mg
12 days is three half-lives, so the mass halves three times: 480 to 240 to 120 to 60 mg. Dividing 480 by 3 rather than halving three times gives 160.
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Which expression is equivalent to (2a^3)^3?
- A6a^9
- B8a^9
- C2a^9
- D8a^6
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Answer: 8a^9
The power of 3 applies to both factors: 2^3 = 8 and (a^3)^3 = a^9. Multiplying the coefficient by the exponent gives 6a^9, and adding the exponents instead of multiplying gives a^6.
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If the square root of (2x + 1) equals 7, what is the value of x?
- A3
- B48
- C24
- D25
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Answer: 24
Squaring gives 2x + 1 = 49, so 2x = 48 and x = 24. The choice 48 stops one step early, and 25 comes from squaring only part of the expression.
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The function f is defined by f(x) = 2x^2 - 3x. Enter the value of f(-3).
Student-produced response: write your own answer rather than choosing one.
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Answer: 27
Substituting gives 2(-3)^2 - 3(-3) = 2(9) + 9 = 18 + 9 = 27. Both signs matter: the square makes the first term positive and subtracting a negative adds.
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The expression 2x^2 + 11x + 12 factors as (2x + 3)(x + k). Enter the value of k.
Student-produced response: write your own answer rather than choosing one.
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Answer: 4
The constants must multiply to 12, so with 3 as one of them the other is 4. Checking the middle term: 2x times 4 gives 8x, and 3 times x gives 3x, and 8x + 3x = 11x.
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Which expression is equivalent to (x - 6)(x + 6)?
- Ax^2 + 36
- Bx^2 - 12x - 36
- Cx^2 - 12x + 36
- Dx^2 - 36
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Answer: x^2 - 36
The middle terms cancel because the constants are opposites: -6x + 6x = 0. This is the difference of squares, and recognising it saves expanding every time.
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What is the value of 16 raised to the power 3/4?
- A8
- B12
- C64
- D4
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Answer: 8
The denominator is the root and the numerator the power: the fourth root of 16 is 2, and 2 cubed is 8. Multiplying 16 by 3/4 gives 12, which treats the exponent as a factor.
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What are the solutions to x^2 - 7x + 12 = 0?
- A-3 and -4
- B3 and 4
- C3 and -4
- D12 and 1
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Answer: 3 and 4
The expression factors as (x - 3)(x - 4). The constants must multiply to 12 and add to -7, which -3 and -4 do, giving solutions 3 and 4.
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A 960 mg sample halves every 5 hours. How much remains after 15 hours?
- A120 mg
- B320 mg
- C480 mg
- D240 mg
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Answer: 120 mg
15 hours is three half-lives, so the mass halves three times: 960 to 480 to 240 to 120 mg. Dividing 960 by 3 gives 320, which halves once in effect rather than three times.
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Which expression is equivalent to (3m^2 n)^2?
- A6m^4 n^2
- B9m^4 n^2
- C9m^4 n
- D3m^4 n^2
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Answer: 9m^4 n^2
The outer square applies to every factor: 3^2 = 9, (m^2)^2 = m^4 and n^2. Multiplying the coefficient by the exponent gives 6, and leaving n unsquared drops one factor.
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If the square root of (5x - 4) equals 6, what is the value of x?
- A32
- B6.4
- C8
- D40
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Answer: 8
Squaring gives 5x - 4 = 36, so 5x = 40 and x = 8. Check: 40 - 4 = 36 and the square root of 36 is 6. The choice 40 stops before the division.
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The function g is defined by g(x) = x^2 + 2x. Enter the value of g(-4).
Student-produced response: write your own answer rather than choosing one.
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Answer: 8
Substituting gives (-4)^2 + 2(-4) = 16 - 8 = 8. The square is positive and the second term is negative, so the two partly cancel.
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The expression 2x^2 + 7x + 3 factors as (2x + 1)(x + k). Enter the value of k.
Student-produced response: write your own answer rather than choosing one.
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Answer: 3
The constants multiply to 3, so with 1 as one of them the other is 3. Checking the middle term: 2x times 3 gives 6x, plus 1 times x, which is 7x.
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Which expression is equivalent to (x - 4)(x + 9)?
- Ax^2 - 5x - 36
- Bx^2 + 5x + 36
- Cx^2 - 36
- Dx^2 + 5x - 36
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Answer: x^2 + 5x - 36
The middle term comes from -4 and +9, which add to +5; the constant is their product, -36. Forgetting the middle term entirely gives x^2 - 36.
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What is the value of 27 raised to the power 2/3?
- A9
- B18
- C6
- D54
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Answer: 9
The denominator is the root and the numerator the power: the cube root of 27 is 3, and 3 squared is 9. Multiplying 27 by 2/3 gives 18, which treats the exponent as a factor.
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What are the solutions to x^2 - 11x + 30 = 0?
- A-5 and -6
- B5 and 6
- C5 and -6
- D30 and 1
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Answer: 5 and 6
The expression factors as (x - 5)(x - 6): the two constants multiply to 30 and add to -11. The solutions are the opposites of the signs in the brackets.
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A sample of 800 mg halves every 4 hours. How much remains after 16 hours?
- A200 mg
- B100 mg
- C50 mg
- D25 mg
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Answer: 50 mg
16 hours is four half-lives, so the mass halves four times: 800 to 400 to 200 to 100 to 50 mg. Dividing 800 by 4 gives 200, which halves once in effect.
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Which expression is equivalent to (4p^3 q^2)^2?
- A8p^6 q^4
- B16p^5 q^4
- C16p^9 q^4
- D16p^6 q^4
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Answer: 16p^6 q^4
The outer square applies to every factor: 4^2 = 16, (p^3)^2 = p^6 and (q^2)^2 = q^4. Adding the exponents rather than multiplying gives p^5.
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If the square root of (4x + 9) equals 5, what is the value of x?
- A4
- B16
- C6.5
- D34
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Answer: 4
Squaring gives 4x + 9 = 25, so 4x = 16 and x = 4. Check: 16 + 9 = 25 and the square root of 25 is 5. The choice 16 stops before dividing by 4.
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The function h is defined by h(x) = x^2 - 5x. Enter the value of h(-3).
Student-produced response: write your own answer rather than choosing one.
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Answer: 24
Substituting gives (-3)^2 - 5(-3) = 9 + 15 = 24. The square is positive and subtracting a negative adds, so both terms are positive.
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The expression 3x^2 + 10x + 8 factors as (3x + 4)(x + k). Enter the value of k.
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Answer: 2
The constants multiply to 8, so with 4 as one of them the other is 2. Checking the middle term: 3x times 2 gives 6x, and 4 times x gives 4x, and 6x + 4x = 10x.
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