complete each quadratic expression to make it a perfect square?

Question

1. Complete each quadratic expression to make it a perfect:
A.
x ^ 2 - 16x + _____

B.
9r ^ 2 + 6r ______

2. Solve each equation by completing the square.
A.
c^2 + 6c - 9 = 0

B.
4m^2 - 12m - 7 = 0

C.
2x^2 - 8x + 5 = 0

3. Explain why b^2 - 6b + 12 = 0 has no solutions.

Answer 1

1a -- x² - 16x + 64 (64 is [16/2]² )
1b -- 9r² + 6r + 1 = (3r + 1)²

2a:
c² + 6c - 9 = 0
c² + 6c + 9 = 18
(c + 3)² = 18
c + 3 = ±√18
c = -3 ± 3√2

2b:
4m² - 12m = 7
4(m² - 3m + 9/4) = 7 + 4(9/4)
4(m - 3/2)² = 16
(m - 3/2)² = 4
m - 3/2 = ±2
m = 3/2 + 2 = 7/2 or
m = 3/2 - 2 = -1/2

2c:
2x² - 8x = -5
2(x² - 4x + 4) = -5 + 2(4)
(x - 2)² = 3
x = 2 ± √3

3:
b² - 6b + 12 = 0
(b² - 6b + 9) = -12 + 9
(b - 3)² = -3
b = 3 ± √-3
b = 3 ± i√3

equation has no REAL solutions because of √-n, but it has 2 complex number solutions.

Answer 2

A. (x-8)^2, so the missing blank = +64
B. (3r+1)^2 = 9r^2 + 6r + 1, so 1 is the missing blank

2A. c^2 + 6c - 9 = 0
c^2 + 6c + (9 - 9) - 9 = 0
c^2 + 6c + 9 - 18 = 0
(c+3)^2 = 18 = 9(2)
c+3 = +-3sqr[2]
c = 3(-1 +- sqr[2])

2B. 4m^2 - 12m - 7 = 0
(2m-3)^2 = 4m^2 - 12m + 9
4m^2 - 12m + (9-9) - 7 = 0
4m^2 - 12m + 9 - 16 = 0
(2m-3)^2 = 16
2m-3 = +-4
2m = 3 +- 4 = 7, -1
m = 7/2, -1/2

2C. 2x^2 - 8x + 5 = 0
x^2 - 4x + 5/2 = 0 y dividing out by 2 to make the first term easier to handle
x^2 - 4x + (4-4) + 5/2 = 0
x^2 - 4x + 4 = 4 - 5/2 = 8/2 - 5/2 = 3/2
(x-2)^2 = 3/2
x-2 = +- sqr[3/2]
x = 2 +- sqr[3/2]

3. Let us try completing the square on this first:
b^2 - 6b + 12 = 0
b^2 - 6b + (9-9) + 12 = 0
b^2 - 6b + 9 = -3
(b-3)^2 = -3
Attempting to take square root of both sides yields the square root of a negative number; hence, no solutions unless you consider complex numbers