Question

20 Points!!

Complete The Square formula for:
x2 -16x + 39 = 0

Answer 1

Please check the explanation.

Explanation:

Given the equation

`x^2\:-16x\:+\:39\:=\:0`

Subtract 39 from both sides

`x^2-16x+39-39=0-39`

Simplify

`x^2-16x=-39`

Rewrite in the form x² + 2ax + a²

solve for a, 2ax = -16x

2ax = -16x

divide both sides by 2a

2ax/2x = -16x/2x

a = -8

Add a² = (-8)² to both sides

`x^2-16x+\left(-8\right)^2=-39+\left(-8\right)^2`

simplify

`x^2-16x+\left(-8\right)^2=25`

Apply perfect square formula: (a-b)² = a² - 2ab + b²

`\left(x-8\right)^2=25` ∵
`x^2-16x+\left(-8\right)^2=\left(x-8\right)^2`

Thus,

`\left(x-8\right)^2=25`

BONUS! EXTENDED SOLUTION!

We can further solve for x such as

`\mathrm{For\:}f^2\left(x\right)=a\mathrm{\:the\:solutions\:are\:}f\left(x\right)=√(a),\:-√(a)`

solve

`x-8=√(25)`

`x-8=√(5^2)`

`x-8=5`

`x=13`

similarly,

`x-8=-√(25)`

`x-8=-5`

`x=3`

Therefore,

`x=13,\:x=3`