Answer
We have to be careful and check the proposed solutions since the solutions might not work in the original equation.
Work Step by Step
Example:
$2x+\sqrt {x+1} =8$
$2x+\sqrt {x+1}-2x =8-2x$
$\sqrt {x+1} = 8-2x$
$(\sqrt {x+1})^2 = (8-2x)^2$
$x+1 = (8-2x)(8-2x)$
$x+1 = 8*8+8*-2x+(-2x)(8)+(-2x)(-2x)$
$x+1 = 64-16x-16x+4x^2$
$x+1 = 64-32x+4x^2$
$x+1 = 4x^2-32x+64$
$x+1-x-1=4x^2-32x+64-x-1$
$0=4x^2-33x+63$
$0=4x^2-21x-12x+63$
$0=x(4x-21)-3(4x-21)$
$0=(x-3)(4x-21)$
$x-3=0$
$x-3+3=0+3$
$x=3$
$4x-21=0$
$4x-21+21=0+21$
$4x=21$
$4x/4=21/4$
$x=5.25$
$x=3$
$2x+\sqrt {x+1} =8$
$2*3+\sqrt {3+1} =8$
$6+\sqrt 4 =8$
$6+2 =8$ (true)
$x=5.25$
$2x+\sqrt {x+1} =8$
$2*5.25+\sqrt {5.25+1} =8$
$10.5+\sqrt {6.25} = 8$
$10.5 + 2.5 =8$
$13 \ne 8$ (not a valid answer)
