Completing the Square - Accuplacer Aaf

The expression $x^2 + 18x + c$ is equivalent to $(x+h)^2$ . What is the value of $c-h$ ?
Answer: 72 For $x^2 + 18x + c$ to be a perfect square $(x+h)^2$ , we must have $2h = 18$ , so $h=9$ . The constant term $c$ must be $h^2$ , so $c = 9^2 = 81$ . The question asks for the value of $c-h$ , which is $81 - 9 = 72$ .
To convert the quadratic function $y = 2x^2 - 20x + 15$ to vertex form, a student's first step is to factor out 2, resulting in $y = 2(x^2 - 10x) + 15$ . What value must be added and subtracted inside the parentheses to complete the square?
Answer: 25 The expression inside the parentheses is $x^2 - 10x$ . To complete the square, take half of the coefficient of the $x$ -term, which is $-10/2 = -5$ . Then, square this result: $(-5)^2 = 25$ . This is the value that must be added inside the parentheses to form a perfect square trinomial. The student would write $y = 2(x^2 - 10x + 25 - 25) + 15$ .
A student attempts to complete the square for $2x^2 + 16x + 18$ and writes $2(x^2 + 8x + 16) + 18 = 2(x + 4)^2 + 18$ . What error did the student make?
Answer: Did not adjust the constant term after adding 16 inside the parentheses The student correctly factored out 2 from the first two terms: 2(x² + 8x) + 18, and correctly identified that 16 should be added to complete the square (since (8/2)² = 16). However, when writing 2(x² + 8x + 16), the student added 2(16) = 32 to the expression but failed to subtract this from the constant term. The correct form should be 2(x² + 8x + 16) + 18 - 32 = 2(x + 4)² - 14. Choice A is wrong because the student did factor correctly initially. Choice B is wrong because 16 is the correct value. Choice D is wrong because (x + 4)² is correct.
What constant must be added to the expression $x^2 - 7x$ to create a perfect square trinomial?
Answer: 49/4 To complete the square for an expression $x^2 + bx$ , we must add the square of half the coefficient of $x$ . In this case, $b = -7$ . Half of -7 is $-7/2$ . Squaring this value gives $(-7/2)^2 = 49/4$ . Adding this constant creates the perfect square trinomial $x^2 - 7x + 49/4$ , which is equal to $(x - 7/2)^2$ .
A student is rewriting the expression $x^2 - 12x + 5$ by completing the square. The result is in the form $(x-h)^2 + k$ . What is the value of $k$ ?
Answer: -31 To complete the square for $x^2 - 12x + 5$ , take half of the coefficient of the $x$ -term (which is -12) to get -6, and then square it: $(-6)^2 = 36$ . Add and subtract 36 to the expression: $(x^2 - 12x + 36) - 36 + 5$ . The part in the parentheses is a perfect square: $(x-6)^2$ . Combining the constants gives $-36 + 5 = -31$ . So the expression is $(x-6)^2 - 31$ . The value of $k$ is -31.
When $-x^2 + 6x + 2$ is written in vertex form by completing the square, which of the following is the result?
Answer: $-(x - 3)^2 + 11$ First factor out the negative coefficient: -(x² - 6x) + 2. To complete the square inside the parentheses, take half of -6 and square it: (-6/2)² = 9. Add and subtract 9: -(x² - 6x + 9 - 9) + 2 = -((x - 3)² - 9) + 2 = -(x - 3)² + 9 + 2 = -(x - 3)² + 11. Choice B uses the wrong sign in the parentheses. Choice C fails to properly distribute the negative when removing -9 from inside the parentheses. Choice D incorrectly uses 6 instead of 3 inside the parentheses.
A quadratic function is rewritten as $y = (x+3.5)^2 - 1.25$ . Which of the following was the original function in standard form?
Answer: $y = x^2 + 7x + 11$ Expand the vertex form $y = (x+3.5)^2 - 1.25$ . The term $(x+3.5)^2$ expands to $x^2 + 2(3.5)x + (3.5)^2 = x^2 + 7x + 12.25$ . Now, subtract the constant: $y = (x^2 + 7x + 12.25) - 1.25$ . This simplifies to $y = x^2 + 7x + 11$ .
If $y = x^2 + 5x - 2$ is rewritten in the form $y = (x+h)^2 + k$ , what is the value of $k$ ?
Answer: -33/4 To complete the square for $x^2 + 5x - 2$ , take half of the $x$ -coefficient (5), which is $5/2$ , and square it: $(5/2)^2 = 25/4$ . Add and subtract this value: $(x^2 + 5x + 25/4) - 25/4 - 2$ . The expression in parentheses is $(x + 5/2)^2$ . The constants are $-25/4 - 2 = -25/4 - 8/4 = -33/4$ . So, the vertex form is $y = (x+5/2)^2 - 33/4$ . The value of $k$ is $-33/4$ .
Which of the following expressions is equivalent to $x^2 - x + 1$ after completing the square?
Answer: $(x - 1/2)^2 + 3/4$ For the expression $x^2 - x + 1$ , the coefficient of the $x$ -term is -1. Half of -1 is $-1/2$ , and squaring this gives $(-1/2)^2 = 1/4$ . Rewrite the expression by adding and subtracting 1/4: $(x^2 - x + 1/4) - 1/4 + 1$ . The terms in the parentheses form a perfect square: $(x - 1/2)^2$ . The constants combine to $-1/4 + 1 = -1/4 + 4/4 = 3/4$ . The equivalent expression is $(x - 1/2)^2 + 3/4$ .
The first step in solving the equation $x^2+20x=4$ by completing the square is to add a constant to both sides. What is that constant?
Answer: 100 To complete the square for the expression $x^2+20x$ , we take half of the coefficient of the $x$ -term (20), which is 10, and square it. $10^2 = 100$ . This is the constant that must be added to both sides of the equation to create a perfect square trinomial on the left side.
By completing the square, the equation $x^2 - 10x = -1$ can be rewritten as $(x-p)^2 = q$ . What is the value of $q$ ?
Answer: 24 The equation is $x^2 - 10x = -1$ . To complete the square on the left side, take half of the $x$ -coefficient (-10), which is -5, and square it: $(-5)^2 = 25$ . Add 25 to both sides of the equation: $x^2 - 10x + 25 = -1 + 25$ . The left side becomes a perfect square: $(x-5)^2 = 24$ . In the form $(x-p)^2 = q$ , we have $p=5$ and $q=24$ . The value of $q$ is 24.
The expression $-x^2 + 8x - 10$ can be written in the vertex form $-(x-h)^2 + k$ . What is the value of $h+k$ ?
Answer: 10 First, factor out -1 from the first two terms: $-(x^2 - 8x) - 10$ . To complete the square, take half of -8 (which is -4) and square it to get 16. Add and subtract 16 inside the parentheses: $-(x^2 - 8x + 16 - 16) - 10$ . This becomes $-((x-4)^2 - 16) - 10$ . Distribute the negative sign: $-(x-4)^2 + 16 - 10$ , which simplifies to $-(x-4)^2 + 6$ . From this form, $h=4$ and $k=6$ . The sum $h+k$ is $4+6 = 10$ .
The expression $x^2 + kx + 12$ is rewritten in the form $(x+h)^2 + 3$ . If $k > 0$ , what is the value of $k$ ?
Answer: 6 Expand the vertex form: $(x+h)^2 + 3 = x^2 + 2hx + h^2 + 3$ . Compare the coefficients of this expanded form to the original expression $x^2 + kx + 12$ . The constant terms must be equal: $h^2 + 3 = 12$ . Solving for $h^2$ gives $h^2 = 9$ , so $h = 3$ or $h = -3$ . The coefficients of the $x$ -term must also be equal: $k = 2h$ . Since the problem states that $k > 0$ , we must have $h > 0$ , so we choose $h=3$ . Therefore, $k = 2(3) = 6$ .
If $-2x^2 + 16x - 25$ is rewritten in the form $a(x-h)^2 + k$ , what is the value of $k$ ?
Answer: 7 First, factor out the coefficient -2 from the first two terms: $-2(x^2 - 8x) - 25$ . To complete the square for $x^2 - 8x$ , we need to add $(-8/2)^2 = (-4)^2 = 16$ inside the parentheses. By doing this, we have effectively subtracted $-2 \times 16 = -32$ . To keep the expression equivalent, we must add 32: $-2(x^2 - 8x + 16) - 25 + 32$ . This simplifies to $-2(x-4)^2 + 7$ . Therefore, the value of $k$ is 7.
The function $f(x) = 2x^2 + 8x - 1$ is rewritten in vertex form $f(x) = a(x-h)^2 + k$ . What is the value of the sum $a+h+k$ ?
Answer: -9 The value of $a$ is the leading coefficient, so $a=2$ . Factor 2 from the first two terms: $2(x^2+4x) - 1$ . Complete the square for $x^2+4x$ by adding $(4/2)^2 = 4$ . We get $2(x^2+4x+4) - 1 - 2(4)$ . This simplifies to $2(x+2)^2 - 1 - 8 = 2(x+2)^2 - 9$ . From this form, $h=-2$ and $k=-9$ . The sum is $a+h+k = 2 + (-2) + (-9) = -9$ .
After completing the square, the expression $x^2 + bx + 15$ becomes $(x + 4)^2 + k$ . What are the values of $b$ and $k$ ?
Answer: $b = 8, k = -1$ When you encounter completing the square problems, you're working backwards from a completed square form to find the original expression's coefficients. To find $b$ and $k$ , expand $(x + 4)^2 + k$ and match it to $x^2 + bx + 15$ . Expanding: $(x + 4)^2 + k = x^2 + 8x + 16 + k$ For this to equal $x^2 + bx + 15$ , the coefficients must match: - The $x$ coefficient: $b = 8$ - The constant term: $16 + k = 15$ , so $k = -1$ Let's examine why the other answers are incorrect: Choice A ( $b = 4, k = -1$ ) uses the number inside the parentheses (4) as the coefficient of $x$ , but when you expand $(x + 4)^2$ , you get $8x$ , not $4x$ . Choice C ( $b = 8, k = 31$ ) correctly identifies $b = 8$ but calculates $k$ incorrectly. This might come from adding instead of subtracting: $16 + 15 = 31$ rather than solving $16 + k = 15$ . Choice D ( $b = 4, k = 31$ ) combines both errors from choices A and C—using 4 as the $x$ -coefficient and miscalculating $k$ . The key strategy here is to always expand the completed square form and match coefficients term by term. Remember that $(x + a)^2 = x^2 + 2ax + a^2$ , so the coefficient of $x$ is always twice the number in the parentheses.
When the expression $4x^2 + 8x - 5$ is written in the form $4(x-h)^2 + k$ , what is the value of $k$ ?
Answer: -9 First, factor out 4 from the first two terms: $4(x^2 + 2x) - 5$ . To complete the square for $x^2 + 2x$ , we add $(2/2)^2 = 1^2 = 1$ inside the parentheses. This gives $4(x^2 + 2x + 1) - 5$ . However, by adding 1 inside the parentheses, we have effectively added $4 \times 1 = 4$ to the expression. To keep it equivalent, we must subtract 4: $4(x^2 + 2x + 1) - 5 - 4$ . This simplifies to $4(x+1)^2 - 9$ . Thus, the value of $k$ is -9.
If the expression $2x^2 + 12x + 8$ is written in the form $a(x-h)^2 + k$ , what is the value of $h$ ?
Answer: -3 First, factor out the leading coefficient, 2, from the first two terms: $2(x^2 + 6x) + 8$ . To complete the square inside the parentheses, take half of the $x$ -coefficient (6), which is 3, and square it to get 9. The expression becomes $2((x+3)^2) - 18 + 8 = 2(x+3)^2 - 10$ . The general vertex form is $a(x-h)^2 + k$ . Comparing $(x-h)^2$ to $(x+3)^2$ , we see that $x-h = x+3$ , which means $-h = 3$ , so $h = -3$ .
Which expression represents the correct first step in completing the square for $4x^2 - 20x + 9$ ?
Answer: $4(x^2 - 5x) + 9$ When completing the square for a quadratic expression, your first step is always to factor out the coefficient of the squared term from the first two terms only. This isolates the variable terms so you can work with a simpler expression inside the parentheses. For $4x^2 - 20x + 9$ , you need to factor out the 4 from just the $x^2$ and $x$ terms. When you factor 4 out of $4x^2 - 20x$ , you get $4(x^2 - 5x)$ . The constant term 9 stays separate, giving you $4(x^2 - 5x) + 9$ . This matches answer choice D. Let's see why the other options are incorrect. Choice A factors out 4 from $4x^2 - 20x$ , but incorrectly writes $4(x^2 - 20x)$ instead of $4(x^2 - 5x)$ . When you factor 4 from $-20x$ , you get $-5x$ , not $-20x$ . Choice B attempts to factor the entire expression at once as $4(x^2 - 5x + 9)$ , but this doesn't equal the original expression since $4 \times 9 = 36$ , not 9. Choice C rewrites $4x^2$ as $(4x)^2$ , which is mathematically correct but doesn't help with completing the square—you need the coefficient outside the parentheses. Remember: when completing the square, always factor out the leading coefficient from only the first two terms. Keep the constant term separate until the final step. This sets you up to easily complete the square inside the parentheses.
A quadratic equation is manipulated by completing the square, resulting in the equation $(x+6)^2 = 10$ . What was the original equation in the form $x^2 + bx + c = 0$ ?
Answer: $x^2 + 12x + 26 = 0$ To find the original equation, expand the completed square form and set it equal to zero. Expand $(x+6)^2$ to get $x^2 + 12x + 36$ . So the equation is $x^2 + 12x + 36 = 10$ . To put it in the form $x^2 + bx + c = 0$ , subtract 10 from both sides: $x^2 + 12x + 36 - 10 = 0$ , which simplifies to $x^2 + 12x + 26 = 0$ .

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