Solve x=sqrt{7x+8}

$$x = \sqrt{ 7x+8 }$$

Solve for x

$$x=8$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(\sqrt{7x+8}\right)^{2}$$

Calculate \sqrt{7x+8} to the power of 2 and get 7x+8.

$$x^{2}=7x+8$$

Subtract 7x from both sides.

$$x^{2}-7x=8$$

Subtract 8 from both sides.

$$x^{2}-7x-8=0$$

To solve the equation, factor x^{2}-7x-8 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-7 ab=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -7.

$$a=-8 b=1$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-8\right)\left(x+1\right)$$

To find equation solutions, solve x-8=0 and x+1=0.

$$x=8 x=-1$$

Substitute 8 for x in the equation x=\sqrt{7x+8}.

$$8=\sqrt{7\times 8+8}$$

Simplify. The value x=8 satisfies the equation.

$$8=8$$

Substitute -1 for x in the equation x=\sqrt{7x+8}.

$$-1=\sqrt{7\left(-1\right)+8}$$

Simplify. The value x=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Equation x=\sqrt{7x+8} has a unique solution.

$$x=8$$