Solve sqrt{x-5}=sqrt{3x+7}-4

$$\sqrt { x - 5 } = \sqrt { 3 x + 7 } - 4$$

Solve for x

$$x=14$$

$$x=6$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x-5}\right)^{2}=\left(\sqrt{3x+7}-4\right)^{2}$$

Calculate \sqrt{x-5} to the power of 2 and get x-5.

$$x-5=\left(\sqrt{3x+7}-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{3x+7}-4\right)^{2}.

$$x-5=\left(\sqrt{3x+7}\right)^{2}-8\sqrt{3x+7}+16$$

Calculate \sqrt{3x+7} to the power of 2 and get 3x+7.

$$x-5=3x+7-8\sqrt{3x+7}+16$$

Add 7 and 16 to get 23.

$$x-5=3x+23-8\sqrt{3x+7}$$

Subtract 3x+23 from both sides of the equation.

$$x-5-\left(3x+23\right)=-8\sqrt{3x+7}$$

To find the opposite of 3x+23, find the opposite of each term.

$$x-5-3x-23=-8\sqrt{3x+7}$$

Combine x and -3x to get -2x.

$$-2x-5-23=-8\sqrt{3x+7}$$

Subtract 23 from -5 to get -28.

$$-2x-28=-8\sqrt{3x+7}$$

Square both sides of the equation.

$$\left(-2x-28\right)^{2}=\left(-8\sqrt{3x+7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-2x-28\right)^{2}.

$$4x^{2}+112x+784=\left(-8\sqrt{3x+7}\right)^{2}$$

Expand \left(-8\sqrt{3x+7}\right)^{2}.

$$4x^{2}+112x+784=\left(-8\right)^{2}\left(\sqrt{3x+7}\right)^{2}$$

Calculate -8 to the power of 2 and get 64.

$$4x^{2}+112x+784=64\left(\sqrt{3x+7}\right)^{2}$$

Calculate \sqrt{3x+7} to the power of 2 and get 3x+7.

$$4x^{2}+112x+784=64\left(3x+7\right)$$

Use the distributive property to multiply 64 by 3x+7.

$$4x^{2}+112x+784=192x+448$$

Subtract 192x from both sides.

$$4x^{2}+112x+784-192x=448$$

Combine 112x and -192x to get -80x.

$$4x^{2}-80x+784=448$$

Subtract 448 from both sides.

$$4x^{2}-80x+784-448=0$$

Subtract 448 from 784 to get 336.

$$4x^{2}-80x+336=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -80 for b, and 336 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 4\times 336}}{2\times 4}$$

Square -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 4\times 336}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-80\right)±\sqrt{6400-16\times 336}}{2\times 4}$$

Multiply -16 times 336.

$$x=\frac{-\left(-80\right)±\sqrt{6400-5376}}{2\times 4}$$

Add 6400 to -5376.

$$x=\frac{-\left(-80\right)±\sqrt{1024}}{2\times 4}$$

Take the square root of 1024.

$$x=\frac{-\left(-80\right)±32}{2\times 4}$$

The opposite of -80 is 80.

$$x=\frac{80±32}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{80±32}{8}$$

Now solve the equation x=\frac{80±32}{8} when ± is plus. Add 80 to 32.

$$x=\frac{112}{8}$$

Divide 112 by 8.

$$x=14$$

Now solve the equation x=\frac{80±32}{8} when ± is minus. Subtract 32 from 80.

$$x=\frac{48}{8}$$

Divide 48 by 8.

$$x=6$$

The equation is now solved.

$$x=14 x=6$$

Substitute 14 for x in the equation \sqrt{x-5}=\sqrt{3x+7}-4.

$$\sqrt{14-5}=\sqrt{3\times 14+7}-4$$

Simplify. The value x=14 satisfies the equation.

$$3=3$$

Substitute 6 for x in the equation \sqrt{x-5}=\sqrt{3x+7}-4.

$$\sqrt{6-5}=\sqrt{3\times 6+7}-4$$

Simplify. The value x=6 satisfies the equation.

$$1=1$$

List all solutions of \sqrt{x-5}=\sqrt{3x+7}-4.

$$x=14 x=6$$