Question

Write x2 - 8x + 13 = 0 in the form (x - a)2 = b, where a and b are integers. a.)(x - 4)2 = 3 b.) (x - 3)2 = 2 c.)(x - 2)2 = 1 d.) (x - 1)2 = 4

Answer 1

The answer is a.)(x - 4)2 = 3.

(x - a)² = b can be expressed as:
x² - 2ax + a² = b                                 ⇒ x² - 2ax + a² - b = 0

Our equation is                x² - 8x + 13 = 0.
The general formula is    x² - 2ax + (a² - b) = 0

Thus:
8x = 2ax    and a² - b = 13

Divide both sides of the first equation (8x = 2ax) by x:
8 = 2a           ⇒ a = 8 ÷ 2 = 4

Replace a in the second equation (a² - b = 13):
4² - b = 13     ⇒ b = 4² - 13 = 16 - 13 = 3

Now when we have a and b, let's just replace them in the general equation:
(x - a)² = b
(x - 4)² = 3