Answer
$x=3$
Work Step by Step
$2x+\sqrt{x+1}=8$
Take the $2x$ to the right side of the equation:
$\sqrt{x+1}=8-2x$
Square both sides of the equation:
$(\sqrt{x+1})^{2}=(8-2x)^{2}$
$x+1=64-2(8)(2x)+4x^{2}$
$x+1=64-32x+4x^{2}$
Take all terms to the right side of the equation:
$0=64-32x+4x^{2}-x-1$
$64-32x+4x^{2}-x-1=0$
Simplify this equation by combining like terms:
$4x^{2}-33x+63=0$
Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For this equation, $a=4$, $b=-33$ and $c=63$
Substitute the known values in the formula:
$x=\dfrac{-(-33)\pm\sqrt{(-33)^{2}-4(4)(63)}}{2(4)}=\dfrac{33\pm\sqrt{1089-1008}}{8}=...$
$...=\dfrac{33\pm\sqrt{81}}{8}=\dfrac{33\pm9}{8}$
Our two preliminary solutions are:
$x=\dfrac{33+9}{8}=\dfrac{21}{4}$
$x=\dfrac{33-9}{8}=3$
Substitute the solutions in the equation to check them:
$2(3)+\sqrt{3+1}=8$
$6+2=8$
$8=8$ True
$2\Big(\dfrac{21}{4}\Big)+\sqrt{\dfrac{21}{4}}=8$
$\dfrac{21+\sqrt{21}}{2}\ne8$ False
The answer is $x=3$
