Solve sqrt{3x+4}-sqrt{x-4}=4

$$\sqrt { 3 x + 4 } - \sqrt { x - 4 } = 4$$

Solve for x

$$x=20$$

$$x=4$$

Solution Steps

Subtract -\sqrt{x-4} from both sides of the equation.

$$\sqrt{3x+4}=4+\sqrt{x-4}$$

Square both sides of the equation.

$$\left(\sqrt{3x+4}\right)^{2}=\left(4+\sqrt{x-4}\right)^{2}$$

Calculate \sqrt{3x+4} to the power of 2 and get 3x+4.

$$3x+4=\left(4+\sqrt{x-4}\right)^{2}$$

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

$$3x+4=16+8\sqrt{x-4}+\left(\sqrt{x-4}\right)^{2}$$

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

$$3x+4=16+8\sqrt{x-4}+x-4$$

Subtract 4 from 16 to get 12.

$$3x+4=12+8\sqrt{x-4}+x$$

Subtract 12+x from both sides of the equation.

$$3x+4-\left(12+x\right)=8\sqrt{x-4}$$

To find the opposite of 12+x, find the opposite of each term.

$$3x+4-12-x=8\sqrt{x-4}$$

Subtract 12 from 4 to get -8.

$$3x-8-x=8\sqrt{x-4}$$

Combine 3x and -x to get 2x.

$$2x-8=8\sqrt{x-4}$$

Square both sides of the equation.

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

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

$$4x^{2}-32x+64=\left(8\sqrt{x-4}\right)^{2}$$

Expand \left(8\sqrt{x-4}\right)^{2}.

$$4x^{2}-32x+64=8^{2}\left(\sqrt{x-4}\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$4x^{2}-32x+64=64\left(\sqrt{x-4}\right)^{2}$$

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

$$4x^{2}-32x+64=64\left(x-4\right)$$

Use the distributive property to multiply 64 by x-4.

$$4x^{2}-32x+64=64x-256$$

Subtract 64x from both sides.

$$4x^{2}-32x+64-64x=-256$$

Combine -32x and -64x to get -96x.

$$4x^{2}-96x+64=-256$$

Add 256 to both sides.

$$4x^{2}-96x+64+256=0$$

Add 64 and 256 to get 320.

$$4x^{2}-96x+320=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -96 for b, and 320 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-96\right)±\sqrt{\left(-96\right)^{2}-4\times 4\times 320}}{2\times 4}$$

Square -96.

$$x=\frac{-\left(-96\right)±\sqrt{9216-4\times 4\times 320}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-96\right)±\sqrt{9216-16\times 320}}{2\times 4}$$

Multiply -16 times 320.

$$x=\frac{-\left(-96\right)±\sqrt{9216-5120}}{2\times 4}$$

Add 9216 to -5120.

$$x=\frac{-\left(-96\right)±\sqrt{4096}}{2\times 4}$$

Take the square root of 4096.

$$x=\frac{-\left(-96\right)±64}{2\times 4}$$

The opposite of -96 is 96.

$$x=\frac{96±64}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{96±64}{8}$$

Now solve the equation x=\frac{96±64}{8} when ± is plus. Add 96 to 64.

$$x=\frac{160}{8}$$

Divide 160 by 8.

$$x=20$$

Now solve the equation x=\frac{96±64}{8} when ± is minus. Subtract 64 from 96.

$$x=\frac{32}{8}$$

Divide 32 by 8.

$$x=4$$

The equation is now solved.

$$x=20 x=4$$

Substitute 20 for x in the equation \sqrt{3x+4}-\sqrt{x-4}=4.

$$\sqrt{3\times 20+4}-\sqrt{20-4}=4$$

Simplify. The value x=20 satisfies the equation.

$$4=4$$

Substitute 4 for x in the equation \sqrt{3x+4}-\sqrt{x-4}=4.

$$\sqrt{3\times 4+4}-\sqrt{4-4}=4$$

Simplify. The value x=4 satisfies the equation.

$$4=4$$

List all solutions of \sqrt{3x+4}=\sqrt{x-4}+4.

$$x=20 x=4$$