# 9th grade algebra (completing square)

posted by Alexis

Can you check the following?
A1. (2x + 3)^2 = 25
sqrt(2x + 3)^2 = +-sqrt(25)
2x + 3 = +- 5
2x + 3 = 5, 2x + 3 = -5
2x = 2, 2x = -8
x = 1, x = -4
{-4, 1}

A2. x^2 + 14x + 45 = 0
x^2 + 14x + __ = -45 + __
x^2 + 14x + 49 = -45 + 49
(x +7)^2 = 4
sqrt(x+7)^2 = +- sqrt(4)
x + 7 = +- 2
x + 7 = 2, x +7 = -2
x = -5, x = -9
{-9, -5}

B2. 3x^2 + 12x - 1 = 14
3x^2 + 12x + __ = 14 + 1
3(x^2 + 4x + __) = 15 + __
3(x^2 + 4x + 4) = 15 + 12
3(x + 2)^2 = 27
(x + 2)^2 = 9
sqrt(x+2)^2 = +- sqrt(9)
x + 2 = +- 3
x + 2 = 3, x + 2 = -3
x = 1, x = -5
{-5, 1}

C1. 3x^2 - 12x - 66 = -3
3x^2 - 12x + __ = -3 + 66
3x^2 - 12x + __ = 63
3(x^2 - 4x + 4) = 63 + 12
3(x - 2)^2 = 75
(x - 2)^2 = 25
sqrt(x-2)^2 = +- sqrt(25)
x - 2 = +- 5
x - 2 = 5, x - 2 = -5
x = 7, x = -3
{-3, 7}

D1. 2x^2 - 12x - 11 = 3
2x^2 - 12x + __ = 3 + 11
2x^2 - 12x + __ = 14 + __
2(x^2 - 6x + 9) = 14 + 18
2(x-3)^2 = 32
(x-3)^2 = 16
sqrt(x-3)2 = +- sqrt(16)
x - 3 = +- 4
x - 3 = 4, x - 3 = -4
x = 7, x = -1
{-1, 7}

D2. (x+4)^2 = 49
sqrt(x+4)^2 = +- sqrt(49)
x + 4 = +- 7
x + 4 = 7, x + 4 = -7
x = 3, x = -11
{-11, 3}

E1. 4x^2 + 16x + 11 = 4
4x^2 + 16x + __ = 4 - 11
4x^2 + 16x + __ = -7 + __
4(x^2 + 4x + 4) = -7 + 16
4(x+2)^2 = 9
(x+2)^2 = 9/4
sqrt(x+2)^2 = +- sqrt(9/4)
x + 2 = +- 3/2
x + 2 = 3/2, x + 2 = -3/2
x = -1/2, x = -7/2
{-7/2, -1/2}

E2. x^2 - 16x = -63
(x^2 - 16x + 64) = -63 + 64
(x - 8)^2 = 1
sqrt(x - 8)^2 = +- sqrt(1)
x - 8 = +- 1
x - 8 = 1, x - 8 = -1
x = 9, x = 7
{7,9}

F2. x^2 + 8x - 63 = 21
x^2 + 8x + __ = 21 + 63
x^2 + 8x + 16 = 84 + 16
(x + 4)^2 = 100
sqrt(x + 4)^2 = +- sqrt(100)
x + 4 = 10, x + 4 = -10
x = 6, x = -14
{-14, 6}

G1. (4x - 1)^2 = 81
sqrt(4x-1)^2 = +- sqrt(81)
4x - 1 = +- 9
4x - 1 = 9, 4x -1 = -9
4x = 10 , 4x = -8
x = 10/4 , 4x = -2
x = 5/2
{-2, 5/2}

G2. x^2 + 10x + 13 = -3
x^2 + 10x + __ = -3 - 13
x^2 + 10x + 25 = -16 + 25
sqrt(x + 5)^2 = +- sqrt(9)
x + 5 = 3, x + 5 = -3
x = -2, x = -8
{-8, -2}

H1. x^2 - 2x + 3 = 27
x^2 - 2x + __ = 27 - 3
x^2 - 2x + 1 = 24 + 1
(x - 1)^2 = 25
sqrt(x - 1)^2 = +- sqrt(25)
x - 1 = +- 5
x - 1 = 5, x - 1 = -5
x = 6, x = -4
{-4, 6}

H2. 9x^2 - 18x - 18 = -2
9x^2 - 18x + __ = - 2 + 18 + __
9(x^2 - 2x + 1) = 16 + 9
9(x-1)^2 = 25
(x-1)^2 = 25/9
sqrt(x-1)^2 = +- sqrt(25/9)
x - 1 = +- 5/3
x - 1 = 5/3, x - 1 = -5/3
x = 8/3, x = -2/3
{-2/3, 8/3}

1. Reiny

You seem to have a good grasp of doing these.
I verified most of your answers by subbing them back into the original equation.
All the ones I tried were correct.

2. Alexis

Thanks for checking. For the ones above, they didn't match any anwers from the homework puzzle. So, I wasn't sure I was doing them correctly.

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