4 = x + 2 sqrt(x-7)4=x+2√x−7
4 -x = 2 sqrt(x-7)4−x=2√x−7
square both sides
(4 -x)^2 = (2 sqrt(x-7))^2(4−x)2=(2√x−7)2
16 - 8 x +x^2 = 4(x-7)16−8x+x2=4(x−7)
16 - 8 x +x^2 = 4x- 2816−8x+x2=4x−28
rearrange the equation,
x^2 - 8x - 4x + 16 + 28 = 0x2−8x−4x+16+28=0
x^2 - 12x + 44 = 0x2−12x+44=0
a = 1, b = -12, c =44a=1,b=−12,c=44
since b^2 - 4ac < 0b2−4ac<0, so there is no real root for this equation.
We use completing a square to solve it.
x^2 - 12x + 44 = 0x2−12x+44=0
(x -6)^2 - (-6)^2 + 44 = 0(x−6)2−(−6)2+44=0
(x -6)^2 - 36 + 44 = 0(x−6)2−36+44=0
(x -6)^2 + 8 = 0(x−6)2+8=0
(x -6)^2 = -8(x−6)2=−8
(x -6) = +-sqrt (-8) = +-sqrt (-1 * 8) = +- i sqrt 8 = +-2i sqrt2(x−6)=±√−8=±√−1⋅8=±i√8=±2i√2
x = 6 +-2i sqrt2x=6±2i√2
