Solve x-sqrt{x+2}=1

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

Solve for x

$$x = \frac{\sqrt{13} + 3}{2} \approx 3.302775638$$

Solution Steps

Subtract x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

Calculate -1 to the power of 2 and get 1.

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

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

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

Use the distributive property to multiply 1 by x+2.

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

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

$$x+2=1-2x+x^{2}$$

Subtract 1 from both sides.

$$x+2-1=-2x+x^{2}$$

Subtract 1 from 2 to get 1.

$$x+1=-2x+x^{2}$$

Add 2x to both sides.

$$x+1+2x=x^{2}$$

Combine x and 2x to get 3x.

$$3x+1=x^{2}$$

Subtract x^{2} from both sides.

$$3x+1-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+3x+1=0$$

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

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

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-3±\sqrt{9+4}}{2\left(-1\right)}$$

Add 9 to 4.

$$x=\frac{-3±\sqrt{13}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-3±\sqrt{13}}{-2}$$

Now solve the equation x=\frac{-3±\sqrt{13}}{-2} when ± is plus. Add -3 to \sqrt{13}.

$$x=\frac{\sqrt{13}-3}{-2}$$

Divide -3+\sqrt{13} by -2.

$$x=\frac{3-\sqrt{13}}{2}$$

Now solve the equation x=\frac{-3±\sqrt{13}}{-2} when ± is minus. Subtract \sqrt{13} from -3.

$$x=\frac{-\sqrt{13}-3}{-2}$$

Divide -3-\sqrt{13} by -2.

$$x=\frac{\sqrt{13}+3}{2}$$

The equation is now solved.

$$x=\frac{3-\sqrt{13}}{2} x=\frac{\sqrt{13}+3}{2}$$

Substitute \frac{3-\sqrt{13}}{2} for x in the equation x-\sqrt{x+2}=1.

$$\frac{3-\sqrt{13}}{2}-\sqrt{\frac{3-\sqrt{13}}{2}+2}=1$$

Simplify. The value x=\frac{3-\sqrt{13}}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$2-13^{\frac{1}{2}}=1$$

Substitute \frac{\sqrt{13}+3}{2} for x in the equation x-\sqrt{x+2}=1.

$$\frac{\sqrt{13}+3}{2}-\sqrt{\frac{\sqrt{13}+3}{2}+2}=1$$

Simplify. The value x=\frac{\sqrt{13}+3}{2} satisfies the equation.

$$1=1$$

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

$$x=\frac{\sqrt{13}+3}{2}$$