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)
$x=5.25$
$2x+\sqrt {x+1}=8$
$2*5.25+\sqrt {5.25+1}=8$
$10.5+\sqrt {6.25}=8$
$10.5+2.5=8$
$13≠8$ (not a valid answer)