Question

How do you solve `1/2(x+8)^2=14`?

Answer 1

`x = 2sqrt(7) - 8`

Explanation:

`1/"2"(x + 8)^2 = 14`

Multiplying by 2 on both sides

`cancel(2) × 1/cancel("2")(x + 8)^2 = 2 × 14`

`(x + 8)^2 = 28`

Square root both side

`sqrt((x + 8)^2` = `sqrt(28)`

`x + 8 = +-2sqrt(7)`

`x = 2sqrt(7) - 8 and x =- 2sqrt(7) - 8`

Answer 2

`x=-8+-2sqrt7`

Explanation:

`1/2 (x+8)^2=14`

I want to expand the squared term, so to do that I'll first multiply both sides by 2:

`1/2 color(red)(xx2) (x+8)^2=14color(red)(xx2)`

`(x+8)^2=28`

`x^2+16x+64=28`

`x^2+16x+64color(red)(-28)=28color(red)(-28)`

`x^2+16x+36=0`

I don't see any easy factors, so let's use the quadratic formula. The general formula is:

`x=(-color(green)b+-sqrt(color(green)b^2-4color(blue)acolor(brown)c))/(2color(blue)a)`

and relates to a trinomial this way:

`color(blue)ax^2+color(green)bx+color(brown)c`

`x=(-16+-sqrt(16^2-4(1)(36)))/(2(1))`

`x=(-16+-sqrt(256-144))/2`

`x=(-16+-sqrt(112))/2=(-16+-sqrt(16xx7))/2=(-16+-4sqrt(7))/2=-8+-2sqrt7`

And we can see these two solutions in the graphing of both sides of the original equation:

graph{(y-(1/2)(x+8)^2)(y-0x-14)=0 [-16.69, 3.31, 9.14, 19.136]}