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

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

Solve for x

$$x = \frac{9 - \sqrt{37}}{2} \approx 1.458618735$$

Solution Steps

Subtract -\sqrt{3x-2} from both sides of the equation.

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

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{3x-2}\right)^{2}.

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

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

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

Subtract 2 from 1 to get -1.

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

Subtract -1+3x from both sides of the equation.

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

To find the opposite of -1+3x, find the opposite of each term.

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

Add 5 and 1 to get 6.

$$x+6-3x=2\sqrt{3x-2}$$

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

$$-2x+6=2\sqrt{3x-2}$$

Square both sides of the equation.

$$\left(-2x+6\right)^{2}=\left(2\sqrt{3x-2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-2x+6\right)^{2}.

$$4x^{2}-24x+36=\left(2\sqrt{3x-2}\right)^{2}$$

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

$$4x^{2}-24x+36=2^{2}\left(\sqrt{3x-2}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}-24x+36=4\left(\sqrt{3x-2}\right)^{2}$$

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

$$4x^{2}-24x+36=4\left(3x-2\right)$$

Use the distributive property to multiply 4 by 3x-2.

$$4x^{2}-24x+36=12x-8$$

Subtract 12x from both sides.

$$4x^{2}-24x+36-12x=-8$$

Combine -24x and -12x to get -36x.

$$4x^{2}-36x+36=-8$$

Add 8 to both sides.

$$4x^{2}-36x+36+8=0$$

Add 36 and 8 to get 44.

$$4x^{2}-36x+44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -36 for b, and 44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 4\times 44}}{2\times 4}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 4\times 44}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-36\right)±\sqrt{1296-16\times 44}}{2\times 4}$$

Multiply -16 times 44.

$$x=\frac{-\left(-36\right)±\sqrt{1296-704}}{2\times 4}$$

Add 1296 to -704.

$$x=\frac{-\left(-36\right)±\sqrt{592}}{2\times 4}$$

Take the square root of 592.

$$x=\frac{-\left(-36\right)±4\sqrt{37}}{2\times 4}$$

The opposite of -36 is 36.

$$x=\frac{36±4\sqrt{37}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{36±4\sqrt{37}}{8}$$

Now solve the equation x=\frac{36±4\sqrt{37}}{8} when ± is plus. Add 36 to 4\sqrt{37}.

$$x=\frac{4\sqrt{37}+36}{8}$$

Divide 36+4\sqrt{37} by 8.

$$x=\frac{\sqrt{37}+9}{2}$$

Now solve the equation x=\frac{36±4\sqrt{37}}{8} when ± is minus. Subtract 4\sqrt{37} from 36.

$$x=\frac{36-4\sqrt{37}}{8}$$

Divide 36-4\sqrt{37} by 8.

$$x=\frac{9-\sqrt{37}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{37}+9}{2} x=\frac{9-\sqrt{37}}{2}$$

Substitute \frac{\sqrt{37}+9}{2} for x in the equation \sqrt{x+5}-\sqrt{3x-2}=1.

$$\sqrt{\frac{\sqrt{37}+9}{2}+5}-\sqrt{3\times \frac{\sqrt{37}+9}{2}-2}=1$$

Simplify. The value x=\frac{\sqrt{37}+9}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Substitute \frac{9-\sqrt{37}}{2} for x in the equation \sqrt{x+5}-\sqrt{3x-2}=1.

$$\sqrt{\frac{9-\sqrt{37}}{2}+5}-\sqrt{3\times \frac{9-\sqrt{37}}{2}-2}=1$$

Simplify. The value x=\frac{9-\sqrt{37}}{2} satisfies the equation.

$$1=1$$

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

$$x=\frac{9-\sqrt{37}}{2}$$