Solve x-8=sqrt{4x}

$$x - 8 = \sqrt { 4 x }$$

Solve for x

$$x=16$$

Solution Steps

Square both sides of the equation.

$$\left(x-8\right)^{2}=\left(\sqrt{4x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-8\right)^{2}.

$$x^{2}-16x+64=\left(\sqrt{4x}\right)^{2}$$

Calculate \sqrt{4x} to the power of 2 and get 4x.

$$x^{2}-16x+64=4x$$

Subtract 4x from both sides.

$$x^{2}-16x+64-4x=0$$

Combine -16x and -4x to get -20x.

$$x^{2}-20x+64=0$$

To solve the equation, factor x^{2}-20x+64 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.

$$a+b=-20 ab=64$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 64.

$$-1,-64 -2,-32 -4,-16 -8,-8$$

Calculate the sum for each pair.

$$-1-64=-65 -2-32=-34 -4-16=-20 -8-8=-16$$

The solution is the pair that gives sum -20.

$$a=-16 b=-4$$

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

$$\left(x-16\right)\left(x-4\right)$$

To find equation solutions, solve x-16=0 and x-4=0.

$$x=16 x=4$$

Substitute 16 for x in the equation x-8=\sqrt{4x}.

$$16-8=\sqrt{4\times 16}$$

Simplify. The value x=16 satisfies the equation.

$$8=8$$

Substitute 4 for x in the equation x-8=\sqrt{4x}.

$$4-8=\sqrt{4\times 4}$$

Simplify. The value x=4 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-4=4$$

Equation x-8=\sqrt{4x} has a unique solution.

$$x=16$$