Solve 4sqrt{x}-4=x

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

Solve for x

$$x=4$$

Solution Steps

Subtract -4 from both sides of the equation.

$$4\sqrt{x}=x+4$$

Square both sides of the equation.

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

Expand \left(4\sqrt{x}\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$16x=x^{2}+8x+16$$

Subtract x^{2} from both sides.

$$16x-x^{2}=8x+16$$

Subtract 8x from both sides.

$$16x-x^{2}-8x=16$$

Combine 16x and -8x to get 8x.

$$8x-x^{2}=16$$

Subtract 16 from both sides.

$$8x-x^{2}-16=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+8x-16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=8 ab=-\left(-16\right)=16$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 16.

$$1,16 2,8 4,4$$

Calculate the sum for each pair.

$$1+16=17 2+8=10 4+4=8$$

The solution is the pair that gives sum 8.

$$a=4 b=4$$

Rewrite -x^{2}+8x-16 as \left(-x^{2}+4x\right)+\left(4x-16\right).

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

Factor out -x in the first and 4 in the second group.

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

Factor out common term x-4 by using distributive property.

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

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

$$x=4 x=4$$

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

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

Simplify. The value x=4 satisfies the equation.

$$4=4$$

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

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

Simplify. The value x=4 satisfies the equation.

$$4=4$$

List all solutions of 4\sqrt{x}=x+4.

$$x=4 x=4$$