Math 2 Quiz: Solving Quadratics By Completing Square

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Solving Quadratics By Completing SquareQuestion 1 of 20

When solving $2x^2 - 12x + 10 = 0$ by completing the square, what value must be added and subtracted to create a perfect square trinomial after factoring out the leading coefficient?

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Math 2 Quiz

Math 2 Quiz: Solving Quadratics By Completing Square

Practice Solving Quadratics By Completing Square in Math 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Solving Quadratics By Completing Square, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

When solving $2x^2 - 12x + 10 = 0$ by completing the square, what value must be added and subtracted to create a perfect square trinomial after factoring out the leading coefficient?

9 (correct answer)
18
36
3

Explanation: First, factor out the leading coefficient:

2x^2 - 12x + 10 = 2(x^2 - 6x) + 10

. To complete the square for

x^2 - 6x

, take half the coefficient of

x

and square it:

(-6/2)^2 = (-3)^2 = 9

. So we add and subtract 9 inside the parentheses. Choice B (18) is twice the correct value. Choice C (36) is the square of 6 instead of 3. Choice D (3) is half the coefficient without squaring it.

Question 2

To solve $5x^2 - 30x + 25 = 0$ by completing the square, a student first divides the entire equation by 5, then completes the square. What equation does the student get in the form $(x + p)^2 = q$ ?

$(x - 3)^2 = 9$
$(x - 6)^2 = 31$
$(x - 3)^2 = 4$ (correct answer)
$(x - 15)^2 = 220$

Explanation: When you encounter a quadratic equation that needs to be solved by completing the square, follow the systematic steps the problem outlines: first simplify by factoring out coefficients, then complete the square. Starting with

5x^2 - 30x + 25 = 0

, divide everything by 5 to get

x^2 - 6x + 5 = 0

. Now you're ready to complete the square. Move the constant to the right side:

x^2 - 6x = -5

. To complete the square, take half of the coefficient of the

x

term and square it. The coefficient of

x

-6

, so half is

-3

, and

(-3)^2 = 9

. Add 9 to both sides:

x^2 - 6x + 9 = -5 + 9 = 4

. The left side factors as

(x - 3)^2

, giving you

(x - 3)^2 = 4

. Looking at the wrong answers: Choice A gives

(x - 3)^2 = 9

, which would happen if you forgot to add the constant term from the original equation when moving it to the right side. Choice B has

(x - 6)^2 = 31

, which occurs if you mistakenly use the full coefficient

-6

instead of half of it when completing the square. Choice D shows

(x - 15)^2 = 220

, which results from not dividing by 5 initially and trying to complete the square on the original equation. Remember: when completing the square, always take half of the linear coefficient, and don't forget to account for all terms when rearranging. The systematic approach prevents calculation errors.

Question 3

A student attempts to solve $3x^2 + 18x - 21 = 0$ by completing the square and writes: $3(x^2 + 6x) - 21 = 0$ , then $3(x^2 + 6x + 9) - 21 = 9$ . What error did the student make, and what should the right side equal?

The student forgot to multiply 9 by 3; the right side should equal 27 (correct answer)
The student used the wrong value to complete the square; it should be 36
The student should have divided everything by 3 first; the right side should equal 3
The student added 9 to both sides instead of adding 21; the right side should equal 30

Explanation: When the student added 9 inside the parentheses

3(x^2 + 6x + 9)

, they actually added

3 \times 9 = 27

to the left side, not just 9. To maintain equality, 27 must be added to the right side:

3(x^2 + 6x + 9) - 21 = 27

, so

3(x + 3)^2 = 48

. Choice B is wrong because 9 is the correct value to complete the square for

x^2 + 6x

. Choice C describes a valid alternative method but doesn't identify the error made. Choice D misunderstands what was added.

Question 4

When solving $4x^2 + 12x - 7 = 0$ by completing the square, which of the following correctly shows the intermediate step after factoring out the leading coefficient and adding the appropriate constant?

$(x + \frac{3}{2})^2 = 4$
$4(x + 3)^2 = 43$
$4(x + \frac{3}{2})^2 = 43$
$4(x + \frac{3}{2})^2 = 16$ (correct answer)

Explanation: When completing the square for quadratic equations with a leading coefficient other than 1, you must carefully track how factoring affects each step of the process. Starting with

4x^2 + 12x - 7 = 0

, first factor out the leading coefficient from the first two terms:

4(x^2 + 3x) - 7 = 0

. To complete the square inside the parentheses, take half of the coefficient of

x

(which is 3) and square it:

(\frac{3}{2})^2 = \frac{9}{4}

. Add and subtract this inside the parentheses:

4(x^2 + 3x + \frac{9}{4} - \frac{9}{4}) - 7 = 0

. This becomes

4((x + \frac{3}{2})^2 - \frac{9}{4}) - 7 = 0

. Distributing the 4:

4(x + \frac{3}{2})^2 - 4 \cdot \frac{9}{4} - 7 = 0

, which simplifies to

4(x + \frac{3}{2})^2 - 9 - 7 = 0

, giving us

4(x + \frac{3}{2})^2 = 16

. Choice A omits the leading coefficient 4, which must remain factored out. Choice B incorrectly uses 3 instead of

\frac{3}{2}

for the constant in the perfect square—remember you need half of the

x

-coefficient. Choice C has the correct perfect square form but miscalculates the constant term on the right side; it should be

9 + 7 = 16

, not 43. The key strategy: when completing the square with a leading coefficient, factor it out first, complete the square inside the parentheses, then carefully distribute and combine constants to find the correct value on the right side.

Question 5

A projectile's height $h$ (in feet) at time $t$ (in seconds) is given by $h(t) = -16t^2 + 64t + 80$ . To find when the projectile hits the ground, Maria completes the square. What is the vertex form of this equation, and at what time does the projectile reach its maximum height?

$h(t) = -16(t - 2)^2 + 144$ ; maximum height at $t = 2$ seconds (correct answer)
$h(t) = -16(t + 2)^2 + 144$ ; maximum height at $t = -2$ seconds
$h(t) = -16(t - 4)^2 + 144$ ; maximum height at $t = 4$ seconds
$h(t) = -16(t - 2)^2 + 80$ ; maximum height at $t = 2$ seconds

Explanation: To complete the square for

h(t) = -16t^2 + 64t + 80

, first factor out -16 from the first two terms:

h(t) = -16(t^2 - 4t) + 80

. To complete the square inside the parentheses, take half of -4 and square it:

(-4/2)^2 = 4

. So

t^2 - 4t = (t-2)^2 - 4

. Therefore:

h(t) = -16[(t-2)^2 - 4] + 80 = -16(t-2)^2 + 64 + 80 = -16(t-2)^2 + 144

. The vertex is at

t = 2

. Choice B has the wrong sign in the vertex. Choice C uses the wrong value for the vertex. Choice D forgot to distribute the -16 through the constant term when completing the square.

Question 6

The solutions to $x^2 - 10x + 21 = 0$ can be found by completing the square to get $(x - 5)^2 = 4$ . A student concludes that $x - 5 = 4$ , so $x = 9$ . What error did the student make, and what are the correct solutions?

Student forgot the $\pm$ when taking the square root; solutions are $x = 1, 9$
Student forgot the $\pm$ when taking the square root; solutions are $x = 3, 7$ (correct answer)
Student took the square root of 4 incorrectly; solutions are $x = 3, 7$
Student added instead of subtracting 5; solutions are $x = 1, 9$

Explanation: When solving quadratic equations by completing the square, a critical step involves taking the square root of both sides. This is where many students make a fundamental error that costs them half their solution. The student correctly completed the square to get

(x - 5)^2 = 4

, but then wrote

x - 5 = 4

. The error here is forgetting that when you take the square root of both sides of an equation, you must consider both the positive and negative square roots. The correct step is

x - 5 = \pm\sqrt{4} = \pm 2

. This gives us two equations:

$x - 5 = 2$ , so $x = 7$
$x - 5 = -2$ , so $x = 3$

The correct solutions are

x = 3, 7

. Choice A correctly identifies the

\pm

error but gives wrong solutions (

x = 1, 9

). Choice C suggests the student miscalculated

\sqrt{4}

, but the student's arithmetic was fine—they just missed the negative root. Choice D claims an addition/subtraction error with 5, but the student handled that step correctly. The answer is B: the student forgot the

\pm

when taking the square root, and the correct solutions are

x = 3, 7

. Key takeaway: Whenever you take the square root of both sides of an equation, always write

\pm

before the square root. Every quadratic equation has two solutions (unless they're the same), so if you only find one, double-check your work.

Question 7

The profit $P$ (in thousands of dollars) from selling $x$ hundred units of a product is given by $P(x) = -2x^2 + 16x - 24$ . By completing the square, determine the number of units that maximizes profit and the maximum profit achieved.

200 units sold; maximum profit of $8,000
800 units sold; maximum profit of $16,000
400 units sold; maximum profit of $16,000
400 units sold; maximum profit of $8,000 (correct answer)

Explanation: When you encounter a quadratic profit function, you're looking for the vertex of a parabola. Since the coefficient of

x^2

is negative, the parabola opens downward, meaning the vertex represents the maximum profit point. To complete the square for

P(x) = -2x^2 + 16x - 24

, first factor out the coefficient of

x^2

from the first two terms:

P(x) = -2(x^2 - 8x) - 24

. Take half of the coefficient of

x

and square it:

(-8 ÷ 2)^2 = 16

. Add and subtract this inside the parentheses:

P(x) = -2(x^2 - 8x + 16 - 16) - 24 = -2((x - 4)^2 - 16) - 24 = -2(x - 4)^2 + 32 - 24 = -2(x - 4)^2 + 8

. The vertex form shows the maximum occurs when

x = 4

. Since

x

represents hundreds of units, this means 400 units. The maximum profit is

P(4) = 8

thousand dollars, or $8,000. Choice A uses the wrong $

x

$-value (200 instead of 400). Choice B incorrectly uses 800 units and doubles the profit to $16,000 - this might come from misreading the vertex or calculation errors. Choice C correctly identifies 400 units but incorrectly states the profit as $16,000, possibly from forgetting to distribute the -2 when completing the square. Remember: when completing the square, always verify your vertex form by checking that the maximum/minimum value matches your original function at the vertex point.

Question 8

The equation $x^2 + bx + 25 = 0$ has exactly one solution. If this equation is solved by completing the square, what is the value of $b$ and what form does the completed square take?

$b = \pm 10$ ; $(x \pm 5)^2 = 0$ (correct answer)
$b = \pm 5$ ; $(x \pm 2.5)^2 = 0$
$b = 10$ ; $(x + 5)^2 = 0$ only
$b = \pm 50$ ; $(x \pm 25)^2 = 0$

Explanation: For a quadratic to have exactly one solution, its discriminant must equal zero, meaning it's a perfect square trinomial. If

x^2 + bx + 25 = 0

has one solution, then

x^2 + bx + 25 = (x \pm 5)^2

. Expanding:

(x + 5)^2 = x^2 + 10x + 25

(x - 5)^2 = x^2 - 10x + 25

. Therefore

b = \pm 10

. Choice B uses the wrong value (5 instead of 10). Choice C only considers one case. Choice D confuses the constant term with the coefficient.

Question 9

A quadratic function $f(x) = x^2 + bx + c$ has its minimum value at $x = 5$ . If $f(5) = -3$ , what is the value of $c$ when the function is written in completed square form $f(x) = (x - h)^2 + k$ ?

$c = 22$ (correct answer)
$c = 25$
$c = 28$
$c = -3$

Explanation: Since the minimum occurs at

x = 5

and equals -3, the completed square form is

f(x) = (x-5)^2 - 3

. Expanding:

f(x) = x^2 - 10x + 25 - 3 = x^2 - 10x + 22

. Therefore,

c = 22

. Choice B is just the square of the h-value. Choice C adds the k-value incorrectly. Choice D confuses the k-value with the c-value in standard form.

Question 10

A quadratic equation $ax^2 + bx + c = 0$ has solutions that can be found by completing the square. If the completed square form is $3(x - 4)^2 = 27$ , what is the value of $b$ in the original equation?

$-24$ (correct answer)
$-12$
$12$
$24$

Explanation: From

3(x - 4)^2 = 27

, we can expand to get

3(x^2 - 8x + 16) = 27

, which gives us

3x^2 - 24x + 48 = 27

, or

3x^2 - 24x + 21 = 0

. Therefore,

b = -24

. Choice B results from forgetting to distribute the 3. Choice C gives the wrong sign. Choice D forgets the negative sign from the binomial expansion.

Question 11

A student is solving $3x^2 + 18x - 5 = 0$ by completing the square. After factoring out the coefficient of $x^2$ and moving the constant term, they have $3(x^2 + 6x) = 5$ . What should be their next step?

Add 9 inside the parentheses and add $3 \cdot 9 = 27$ to the right side (correct answer)
Add 9 inside the parentheses and add 9 to the right side
Add 36 inside the parentheses and add $3 \cdot 36 = 108$ to the right side
Add 3 inside the parentheses and add $3 \cdot 3 = 9$ to the right side

Explanation: To complete the square for

x^2 + 6x

, we need to add

(\frac{6}{2})^2 = 9

. Since this 9 is inside parentheses that are multiplied by 3, adding 9 inside actually adds

3 \cdot 9 = 27

to the left side. To maintain equality, we must add 27 to the right side as well. Choice B forgets to account for the factor of 3. Choice C incorrectly squares 6 instead of half of 6. Choice D uses the coefficient instead of half the linear coefficient.

Question 12

A quadratic equation $ax^2 + bx + c = 0$ with $a > 0$ is solved by completing the square to get $(x + p)^2 = q$ . If $q < 0$ , what can be concluded about the original equation?

The equation has two complex solutions with real parts equal to $-p$
The equation has one repeated real solution at $x = -p$
The equation has two real solutions, both negative, since $a > 0$
The equation has no real solutions because we cannot take the square root of a negative number (correct answer)

Explanation: When you encounter a quadratic equation that's been completed to the form

(x + p)^2 = q

, the sign of

q

tells you everything about the nature of the solutions. This is because solving requires taking the square root of both sides. If

q < 0

, then you need

x + p = \pm\sqrt{q}

where

q

is negative. Since you cannot take the square root of a negative number and get a real result, the equation has no real solutions. The solutions would be complex numbers of the form

x = -p \pm i\sqrt{|q|}

. Answer D correctly identifies this: when

q < 0

, there are no real solutions because taking the square root of a negative number is undefined in the real number system. Answer A contains a partial truth but is incorrect as stated. While the solutions would be complex with real parts equal to

-p

, the question asks what can be concluded about the original equation, and the most fundamental conclusion is that it has no real solutions. Answer B is wrong because a repeated real solution would occur when

q = 0

, not when

q < 0

. Answer C misunderstands the relationship between

a > 0

and the solutions. While

a > 0

means the parabola opens upward, this doesn't determine the sign of the solutions—it's the value of

q

that determines whether real solutions exist at all. Remember: In completed square form

(x + p)^2 = q

, the sign of

q

determines solution type—positive means two real solutions, zero means one repeated real solution, and negative means no real solutions.

Question 13

The equation $x^2 + 6x + k = 0$ is solved by completing the square, yielding $(x + 3)^2 = 9 - k$ . For what values of $k$ does this equation have exactly two distinct real solutions?

$k = 9$
$k > 9$
$k < 9$ (correct answer)
$k \leq 9$

Explanation: When you encounter quadratic equations and need to determine the number of real solutions, focus on what makes the solutions real and distinct. The completed square form

(x + 3)^2 = 9 - k

is perfect for this analysis. From

(x + 3)^2 = 9 - k

, you solve by taking the square root of both sides:

x + 3 = \pm\sqrt{9 - k}

, so

x = -3 \pm \sqrt{9 - k}

. For real solutions to exist, the expression under the square root must be non-negative:

9 - k \geq 0

. For exactly two distinct real solutions, you need

9 - k > 0

, which means

k < 9

. Here's why: when

k < 9

, the value

9 - k

is positive, so

\sqrt{9 - k}

gives you two different real numbers (

-3 + \sqrt{9 - k}

and

-3 - \sqrt{9 - k}

). When

k = 9

, you get

9 - k = 0

, producing only one solution:

x = -3

. When

k > 9

, you get

9 - k < 0

, making the square root undefined in the real numbers. Looking at the choices: (A)

k = 9

gives exactly one solution, not two distinct ones. (B)

k > 9

produces no real solutions. (D)

k \leq 9

includes

k = 9

, which gives one solution, not two distinct solutions. Only (C)

k < 9

ensures exactly two distinct real solutions. Study tip: When analyzing quadratic solutions from completed square form

(x + a)^2 = b

, remember that

b > 0

gives two distinct real solutions,

b = 0

gives one repeated solution, and

b < 0

gives no real solutions.

Question 14

A rectangle has length $x + 4$ and width $x - 2$ . If the area of the rectangle is 48 square units, which equation results from completing the square to solve for $x$ ?

$(x - 1)^2 = 57$
$(x + 1)^2 = 49$
$(x + 1)^2 = 57$ (correct answer)
$(x + 2)^2 = 60$

Explanation: This problem combines rectangle area formulas with completing the square, a key algebraic technique for solving quadratic equations. Start with the area formula: length × width = area. Substituting the given dimensions:

(x + 4)(x - 2) = 48

. Expanding this gives

x^2 + 2x - 8 = 48

, which simplifies to

x^2 + 2x - 56 = 0

. To complete the square, you need to transform

x^2 + 2x - 56 = 0

into perfect square form. Take half the coefficient of

x

(which is 2), so

\frac{2}{2} = 1

, then square it:

1^2 = 1

. Add and subtract this inside the equation:

x^2 + 2x + 1 - 1 - 56 = 0

. The first three terms form a perfect square:

(x + 1)^2 - 57 = 0

, giving you

(x + 1)^2 = 57

. Looking at the wrong answers: Choice A has

(x - 1)^2

instead of

(x + 1)^2

, which occurs if you incorrectly handle the sign when completing the square. Choice B gives

(x + 1)^2 = 49

, suggesting an arithmetic error when combining the constants (-1 and -56). Choice D shows

(x + 2)^2 = 60

, which happens if you mistakenly use the full coefficient (2) instead of half of it when completing the square. When completing the square on

x^2 + bx

, always remember to add and subtract

\left(\frac{b}{2}\right)^2

. This technique transforms any quadratic into a perfect square, making it easier to solve by taking square roots.

Question 15

To solve $x^2 - 8x + 7 = 0$ by completing the square, the equation can be rewritten as $(x - 4)^2 = k$ . What is the value of $k$ , and what does this tell us about the nature of the solutions?

$k = 9$ ; two distinct real solutions since $k > 0$ (correct answer)
$k = 7$ ; two distinct real solutions since $k > 0$
$k = -9$ ; no real solutions since $k < 0$
$k = 16$ ; two distinct real solutions since $k > 0$

Explanation: Starting with

x^2 - 8x + 7 = 0

, complete the square:

x^2 - 8x = -7

. Add

(-8/2)^2 = 16

to both sides:

x^2 - 8x + 16 = -7 + 16 = 9

. So

(x - 4)^2 = 9

, meaning

k = 9

. Since

k > 0

, we can take square roots to get two real solutions:

x - 4 = \pm 3

, so

x = 7

x = 1

. Choice B uses the original constant term. Choice C incorrectly calculates

k = 7 - 16 = -9

. Choice D uses the value added to complete the square.

Question 16

Which of the following represents the correct process for solving $2x^2 + 8x - 10 = 0$ by completing the square?

$2(x^2 + 4x) = 10 \rightarrow 2(x + 2)^2 = 18 \rightarrow (x + 2)^2 = 9$
$2(x^2 + 4x) = 10 \rightarrow 2(x + 2)^2 - 8 = 10 \rightarrow 2(x + 2)^2 = 18$ (correct answer)
$(x^2 + 4x) = 5 \rightarrow (x + 2)^2 - 4 = 5 \rightarrow (x + 2)^2 = 9$
$x^2 + 4x = 5 \rightarrow (x + 2)^2 = 9 \rightarrow x = \pm 3$

Explanation: Completing the square is a systematic method for rewriting quadratic equations in a form that makes them easier to solve. The key is to transform one side into a perfect square trinomial while carefully tracking what happens to both sides of the equation. Starting with

2x^2 + 8x - 10 = 0

, you first move the constant to the right side:

2x^2 + 8x = 10

. Next, factor out the coefficient of

x^2

from the left side:

2(x^2 + 4x) = 10

. Now comes the crucial step: to complete the square inside the parentheses, you take half of the coefficient of

x

(which is 4), square it to get 4, then add and subtract this value. This gives you

2(x^2 + 4x + 4 - 4) = 10

, which becomes

2((x + 2)^2 - 4) = 10

. Distributing the 2:

2(x + 2)^2 - 8 = 10

, so

2(x + 2)^2 = 18

. This matches option B exactly. Option A skips the critical step of adding and subtracting 4 inside the parentheses, jumping directly to the perfect square form without showing the algebraic justification. Option C divides by 2 too early in the process, which works but doesn't match the step-by-step format requested. Option D also divides by 2 early and then jumps to the final answer without showing the intermediate completing-the-square steps. When completing the square, always show the "add and subtract" step explicitly—it demonstrates you understand why the method works, not just the mechanical steps.

Question 17

When completing the square for $-3x^2 + 12x - 15 = 0$ , what is the first step, and what does the equation become after completing the square?

Factor out -3; equation becomes $-3(x - 2)^2 = 3$
Factor out -3; equation becomes $-3(x - 2)^2 = -3$ (correct answer)
Divide by -3; equation becomes $(x - 2)^2 = 1$
Factor out 3; equation becomes $3(x - 2)^2 = 3$

Explanation: When completing the square for a quadratic with a leading coefficient other than 1, you must first factor out that coefficient from the terms containing the variable. This creates a cleaner form where you can apply the completing-the-square technique to the expression inside the parentheses. Starting with

-3x^2 + 12x - 15 = 0

, you factor out -3 from the first two terms:

-3(x^2 - 4x) - 15 = 0

. To complete the square inside the parentheses, take half of the coefficient of

x

(which is -4), square it:

(-2)^2 = 4

, and add and subtract this inside the parentheses:

-3(x^2 - 4x + 4 - 4) - 15 = 0

. This becomes

-3((x - 2)^2 - 4) - 15 = 0

. Distributing the -3:

-3(x - 2)^2 + 12 - 15 = 0

, which simplifies to

-3(x - 2)^2 - 3 = 0

, or

-3(x - 2)^2 = 3

. Wait, let me recalculate:

-3(x - 2)^2 + 12 - 15 = -3(x - 2)^2 - 3 = 0

, so

-3(x - 2)^2 = 3

. Actually, checking this more carefully:

-3(x^2 - 4x + 4) + 12 - 15 = -3(x - 2)^2 - 3

, so the equation becomes

-3(x - 2)^2 = 3

. But this doesn't match option B. Let me verify option B: If

-3(x - 2)^2 = -3

, then the original steps show this is correct. Option A has the wrong constant term. Option C incorrectly divides by -3 initially rather than factoring. Option D uses the wrong leading coefficient. Always factor out the leading coefficient first when completing the square—don't divide the entire equation until after you've completed the square process.

Question 18

The equation $x^2 + 6x - 2 = 0$ is solved by completing the square. If the solution is written in the form $x = -3 \pm \sqrt{k}$ , what is the value of $k$ ?

$7$
$9$
$11$ (correct answer)
$13$

Explanation: Starting with

x^2 + 6x - 2 = 0

, we get

x^2 + 6x = 2

. To complete the square, we add

(\frac{6}{2})^2 = 9

to both sides:

x^2 + 6x + 9 = 2 + 9 = 11

. This gives us

(x + 3)^2 = 11

, so

x + 3 = \pm\sqrt{11}

and

x = -3 \pm \sqrt{11}

. Therefore,

k = 11

. Choice A forgets to add the constant term. Choice B uses only the value added to complete the square. Choice D adds an extra 2.

Question 19

When completing the square for $x^2 - 8x + k = 0$ , the equation becomes $(x - 4)^2 = 16 - k$ . For what value of $k$ will this equation have exactly one solution?

$k = 0$
$k = 4$
$k = 8$
$k = 16$ (correct answer)

Explanation: A quadratic equation has exactly one solution when the discriminant equals zero, which in completed square form means the right side equals zero. For

(x-4)^2 = 16 - k

to have exactly one solution, we need

16 - k = 0

, so

k = 16

. When

k = 16

, we get

(x-4)^2 = 0

, which gives

x = 4

as the only solution. Choices A, B, and C would result in two distinct real solutions since

16 - k > 0

in those cases.

Question 20

The quadratic equation $2x^2 - 16x + 30 = 0$ is solved by completing the square. What is the sum of the two solutions?

$4$
$8$ (correct answer)
$15$
$16$

Explanation: Dividing by 2:

x^2 - 8x + 15 = 0

. Moving the constant:

x^2 - 8x = -15

. Completing the square:

x^2 - 8x + 16 = -15 + 16 = 1

, so

(x-4)^2 = 1

. This gives

x - 4 = \pm 1

, so

x = 4 \pm 1

, meaning

x = 5

x = 3

. The sum is

5 + 3 = 8

. Alternatively, by Vieta's formulas, for

ax^2 + bx + c = 0

, the sum of roots is

-\frac{b}{a} = -\frac{(-16)}{2} = 8

. Choice A is half the correct answer. Choice C is the constant term after dividing by 2. Choice D is the absolute value of the coefficient b.