Solve sqrt{3x-1}-sqrt{x+5}=2

$$\sqrt{ 3x-1 } - \sqrt{ x+5 } =2$$

Solve for x

$$x=2\sqrt{11}+7\approx 13.633249581$$

Solution Steps

Subtract -\sqrt{x+5} from both sides of the equation.

$$\sqrt{3x-1}=2+\sqrt{x+5}$$

Square both sides of the equation.

$$\left(\sqrt{3x-1}\right)^{2}=\left(2+\sqrt{x+5}\right)^{2}$$

Calculate \sqrt{3x-1} to the power of 2 and get 3x-1.

$$3x-1=\left(2+\sqrt{x+5}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+\sqrt{x+5}\right)^{2}.

$$3x-1=4+4\sqrt{x+5}+\left(\sqrt{x+5}\right)^{2}$$

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

$$3x-1=4+4\sqrt{x+5}+x+5$$

Add 4 and 5 to get 9.

$$3x-1=9+4\sqrt{x+5}+x$$

Subtract 9+x from both sides of the equation.

$$3x-1-\left(9+x\right)=4\sqrt{x+5}$$

To find the opposite of 9+x, find the opposite of each term.

$$3x-1-9-x=4\sqrt{x+5}$$

Subtract 9 from -1 to get -10.

$$3x-10-x=4\sqrt{x+5}$$

Combine 3x and -x to get 2x.

$$2x-10=4\sqrt{x+5}$$

Square both sides of the equation.

$$\left(2x-10\right)^{2}=\left(4\sqrt{x+5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-10\right)^{2}.

$$4x^{2}-40x+100=\left(4\sqrt{x+5}\right)^{2}$$

Expand \left(4\sqrt{x+5}\right)^{2}.

$$4x^{2}-40x+100=4^{2}\left(\sqrt{x+5}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$4x^{2}-40x+100=16\left(\sqrt{x+5}\right)^{2}$$

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

$$4x^{2}-40x+100=16\left(x+5\right)$$

Use the distributive property to multiply 16 by x+5.

$$4x^{2}-40x+100=16x+80$$

Subtract 16x from both sides.

$$4x^{2}-40x+100-16x=80$$

Combine -40x and -16x to get -56x.

$$4x^{2}-56x+100=80$$

Subtract 80 from both sides.

$$4x^{2}-56x+100-80=0$$

Subtract 80 from 100 to get 20.

$$4x^{2}-56x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -56 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 4\times 20}}{2\times 4}$$

Square -56.

$$x=\frac{-\left(-56\right)±\sqrt{3136-4\times 4\times 20}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-56\right)±\sqrt{3136-16\times 20}}{2\times 4}$$

Multiply -16 times 20.

$$x=\frac{-\left(-56\right)±\sqrt{3136-320}}{2\times 4}$$

Add 3136 to -320.

$$x=\frac{-\left(-56\right)±\sqrt{2816}}{2\times 4}$$

Take the square root of 2816.

$$x=\frac{-\left(-56\right)±16\sqrt{11}}{2\times 4}$$

The opposite of -56 is 56.

$$x=\frac{56±16\sqrt{11}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{56±16\sqrt{11}}{8}$$

Now solve the equation x=\frac{56±16\sqrt{11}}{8} when ± is plus. Add 56 to 16\sqrt{11}.

$$x=\frac{16\sqrt{11}+56}{8}$$

Divide 56+16\sqrt{11} by 8.

$$x=2\sqrt{11}+7$$

Now solve the equation x=\frac{56±16\sqrt{11}}{8} when ± is minus. Subtract 16\sqrt{11} from 56.

$$x=\frac{56-16\sqrt{11}}{8}$$

Divide 56-16\sqrt{11} by 8.

$$x=7-2\sqrt{11}$$

The equation is now solved.

$$x=2\sqrt{11}+7 x=7-2\sqrt{11}$$

Substitute 2\sqrt{11}+7 for x in the equation \sqrt{3x-1}-\sqrt{x+5}=2.

$$\sqrt{3\left(2\sqrt{11}+7\right)-1}-\sqrt{2\sqrt{11}+7+5}=2$$

Simplify. The value x=2\sqrt{11}+7 satisfies the equation.

$$2=2$$

Substitute 7-2\sqrt{11} for x in the equation \sqrt{3x-1}-\sqrt{x+5}=2.

$$\sqrt{3\left(7-2\sqrt{11}\right)-1}-\sqrt{7-2\sqrt{11}+5}=2$$

Simplify. The value x=7-2\sqrt{11} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

Substitute 2\sqrt{11}+7 for x in the equation \sqrt{3x-1}-\sqrt{x+5}=2.

$$\sqrt{3\left(2\sqrt{11}+7\right)-1}-\sqrt{2\sqrt{11}+7+5}=2$$

Simplify. The value x=2\sqrt{11}+7 satisfies the equation.

$$2=2$$

Equation \sqrt{3x-1}=\sqrt{x+5}+2 has a unique solution.

$$x=2\sqrt{11}+7$$