Solve x-2=sqrt{2x+7}

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

Solve for x

$$x=2\sqrt{3}+3\approx 6.464101615$$

Solution Steps

Square both sides of the equation.

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

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

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

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

$$x^{2}-4x+4=2x+7$$

Subtract 2x from both sides.

$$x^{2}-4x+4-2x=7$$

Combine -4x and -2x to get -6x.

$$x^{2}-6x+4=7$$

Subtract 7 from both sides.

$$x^{2}-6x+4-7=0$$

Subtract 7 from 4 to get -3.

$$x^{2}-6x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-3\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-3\right)}}{2}$$

Multiply -4 times -3.

$$x=\frac{-\left(-6\right)±\sqrt{36+12}}{2}$$

Add 36 to 12.

$$x=\frac{-\left(-6\right)±\sqrt{48}}{2}$$

Take the square root of 48.

$$x=\frac{-\left(-6\right)±4\sqrt{3}}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±4\sqrt{3}}{2}$$

Now solve the equation x=\frac{6±4\sqrt{3}}{2} when ± is plus. Add 6 to 4\sqrt{3}.

$$x=\frac{4\sqrt{3}+6}{2}$$

Divide 6+4\sqrt{3} by 2.

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

Now solve the equation x=\frac{6±4\sqrt{3}}{2} when ± is minus. Subtract 4\sqrt{3} from 6.

$$x=\frac{6-4\sqrt{3}}{2}$$

Divide 6-4\sqrt{3} by 2.

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

The equation is now solved.

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

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

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

Simplify. The value x=2\sqrt{3}+3 satisfies the equation.

$$2\times 3^{\frac{1}{2}}+1=2\times 3^{\frac{1}{2}}+1$$

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

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

Simplify. The value x=3-2\sqrt{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$1-2\times 3^{\frac{1}{2}}=2\times 3^{\frac{1}{2}}-1$$

Equation x-2=\sqrt{2x+7} has a unique solution.

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