Question

Which of the following are solutions to the equation below?

Check all that apply.
x2 + 10x + 25 = 8

Answer 1

The correct options for the solution values are:

• 
  `x=2√(2)-5`

• 
  `x=-2√(2)-5`

Explanation:

Given the expression

`x^2+10x+25=8`

Subtract 25 from both sides

`x^2+10x+25-25=8-25`

Simplify

`x^2+10x=-17`

Add 25 or 5² to both sides

`x^2+10x+5^2=8`

as

`x^2+10x+5^2=\left(x+5\right)^2`

so the expression becomes

`\left(x+5\right)^2=8`

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

solve

`x+5=√(8)`

Subtract 5 from both sides

`x+5-5=2√(2)-5`

`x=2√(2)-5`

solve

`x+5=-√(8)`

Subtract 5 from both sides

`x+5-5=-2√(2)-5`

`x=-2√(2)-5`

Therefore, the solution to the equation

`x=2√(2)-5,\:x=-2√(2)-5`

Hence, the correct options for the solution values are:

• 
  `x=2√(2)-5`

• 
  `x=-2√(2)-5`