Solve 4x-4sqrt{x}=8

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

Solve for x

$$x=4$$

Solution Steps

Subtract 4x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

Calculate -4 to the power of 2 and get 16.

$$16\left(\sqrt{x}\right)^{2}=\left(8-4x\right)^{2}$$

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

$$16x=\left(8-4x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-4x\right)^{2}.

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

Subtract 64 from both sides.

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

Add 64x to both sides.

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

Combine 16x and 64x to get 80x.

$$80x-64=16x^{2}$$

Subtract 16x^{2} from both sides.

$$80x-64-16x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-16x^{2}+80x-64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 80 for b, and -64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}-4\left(-16\right)\left(-64\right)}}{2\left(-16\right)}$$

Square 80.

$$x=\frac{-80±\sqrt{6400-4\left(-16\right)\left(-64\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-80±\sqrt{6400+64\left(-64\right)}}{2\left(-16\right)}$$

Multiply 64 times -64.

$$x=\frac{-80±\sqrt{6400-4096}}{2\left(-16\right)}$$

Add 6400 to -4096.

$$x=\frac{-80±\sqrt{2304}}{2\left(-16\right)}$$

Take the square root of 2304.

$$x=\frac{-80±48}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-80±48}{-32}$$

Now solve the equation x=\frac{-80±48}{-32} when ± is plus. Add -80 to 48.

$$x=-\frac{32}{-32}$$

Divide -32 by -32.

$$x=1$$

Now solve the equation x=\frac{-80±48}{-32} when ± is minus. Subtract 48 from -80.

$$x=-\frac{128}{-32}$$

Divide -128 by -32.

$$x=4$$

The equation is now solved.

$$x=1 x=4$$

Substitute 1 for x in the equation 4x-4\sqrt{x}=8.

$$4\times 1-4\sqrt{1}=8$$

Simplify. The value x=1 does not satisfy the equation.

$$0=8$$

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

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

Simplify. The value x=4 satisfies the equation.

$$8=8$$

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

$$x=4$$