16. Completing the square a = 1

Question 1

LEVEL 8

Complete the square for the following quadratic:

x^2 + 4x + 7.

Select the correct answer from the options below:

A: (x + 2)^2 + 3

B: (x + 4)^2 + 7

C: (x + 2)^2- 3

D: (x + 2)^2- 7

CORRECT ANSWER:  A: (x + 2)^2 + 3

WORKED SOLUTION:

To complete the square on this quadratic, we aim to write it in the form

(x + a)^2 + b,

where a and b are numbers to be found.

To find a, we simply half the coefficient of x, so a = 2.

Then to find b, we subtract a^2 = 2^2 = 4 from the constant term, 7,

so b = 3. Therefore, the correct result of completing the square is

Question 2

LEVEL 8

Complete the square for the following quadratic:

t^2 - 6t + 5

Select the correct answer from the options below:

A: (t - 3)^2 - 4

B: (t + 3)^2 + 13

C: (t - 3)^2 + 13

D: (t + 3)^2 - 4

CORRECT ANSWER:   A: (t - 3)^2 - 4

WORKED SOLUTION:

To complete the square on this quadratic, we aim to write it in the form

(t + a)^2 + b,

where a and b are numbers to be found.

To find a, we simply half the coefficient of t, so a = -3.

Then to find b, we subtract a^2 = (-3)^2 = 9 from the constant term, 5,

so b = -4. Therefore, the correct result of completing the square is

(t - 3)^2 - 4

Question 3

LEVEL 8

Complete the square for the following quadratic:

x^2 + 3x - 8

Select the correct answer from the options below:

A: \left(x + \frac{3}{2}\right)^2 - \frac{23}{4}

B: \left(x+ \frac{3}{2}\right)^2 - \frac{41}{4}

C: \left(x + \frac{3}{2}\right)^2 + \frac{23}{4}

D: \left(x - \frac{3}{2}\right)^2 + \frac{41}{4}

CORRECT ANSWER:  B: \left(x+ \frac{3}{2}\right)^2 - \frac{41}{4}

WORKED SOLUTION:

To complete the square on this quadratic, we aim to write it in the form

(x + a)^2 + b,

where a and b are numbers to be found.

To find a, we simply half the coefficient of t, so a = \frac{3}{2}.

Then to find b, we subtract a^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} from the constant term, -8,

so b = -8 - \frac{9}{4} = -\frac{32}{4} - \frac{9}{4} = -\frac{41}{4}.

Therefore, the correct result of completing the square is

\left(x+ \frac{3}{2}\right)^2 - \frac{41}{4}

Question 4

LEVEL 8

Complete the square for the following quadratic:

d^2 - d + 15

Select the correct answer from the options below:

A: \left(d + \frac{3}{2}\right)^2 - \frac{61}{4}

B: \left(x - \frac{3}{2}\right)^2 - \frac{61}{4}

C: \left(d + \frac{1}{2}\right)^2 + \frac{61}{4}

D: \left(d - \frac{1}{2}\right)^2 + \frac{59}{4}

CORRECT ANSWER:      D: \left(d - \frac{1}{2}\right)^2 + \frac{59}{4}

WORKED SOLUTION:

To complete the square on this quadratic, we aim to write it in the form

(d + a)^2 + b,

where a and b are numbers to be found.

To find a, we simply half the coefficient of d, so a = -\frac{1}{2}.

Then to find b, we subtract a^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4} from the constant term, 15,

so b = 15 - \frac{1}{4} = \frac{60}{4} - \frac{1}{4} = \frac{59}{4}.

Therefore, the correct result of completing the square is

Question 5

LEVEL 8

Complete the square for the following quadratic:

x^2 – 4x + 3

Select the correct answer from the options below:

A: (x-2)^2  -1

B: (x-1)^2  -2

C: (x+2)^2  -1

D: (x-2)^2  +1

CORRECT ANSWER:   B: (x-2)^2  -1

WORKED SOLUTION:

To complete the square on this quadratic, we aim to write it in the form

(x + a)^2 + b,

where a and b are numbers to be found.

To find a, we simply half the coefficient of x, so a = -2.

Then to find b, we subtract a^2 = (-2)^2 = 4 from the constant term, 3,

so b = -1. Therefore, the correct result of completing the square is

(x-2)^2  -1

Question 6

LEVEL 8

Complete the square for the following quadratic:

x^2 + 10x + 24 = 0

Select the correct answer from the options below:

A: (x - 5)^2 – 1 = 0

B: (x + 5)^2 +1 = 0

C: (x + 5)^2 – 1 = 0

D: (x + 5)^2 - 2 = 0

CORRECT ANSWER:    C: (x + 5)^2 – 1 = 0

WORKED SOLUTION:

To complete the square on this quadratic, we aim to write it in the form

(x + a)^2 + b,

where a and b are numbers to be found.

To find a, we simply half the coefficient of x, so a = -5.

Then to find b, we subtract a^2 = (-5)^2 = 25 from the constant term, 24,

so b = -1. Therefore, the correct result of completing the square is

(x + 5)^2 - 1 = 0