Solve sqrt{x+2}=x+1

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

Solve for x

$$x=\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Subtract x^{2} from both sides.

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

Subtract 2x from both sides.

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

Combine x and -2x to get -x.

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

Subtract 1 from both sides.

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

Subtract 1 from 2 to get 1.

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

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.

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

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

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

Multiply -4 times -1.

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

Add 1 to 4.

$$x=\frac{-\left(-1\right)±\sqrt{5}}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{1±\sqrt{5}}{-2}$$

Now solve the equation x=\frac{1±\sqrt{5}}{-2} when ± is plus. Add 1 to \sqrt{5}.

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

Divide 1+\sqrt{5} by -2.

$$x=\frac{-\sqrt{5}-1}{2}$$

Now solve the equation x=\frac{1±\sqrt{5}}{-2} when ± is minus. Subtract \sqrt{5} from 1.

$$x=\frac{1-\sqrt{5}}{-2}$$

Divide 1-\sqrt{5} by -2.

$$x=\frac{\sqrt{5}-1}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{5}-1}{2} x=\frac{\sqrt{5}-1}{2}$$

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

$$\sqrt{\frac{-\sqrt{5}-1}{2}+2}=\frac{-\sqrt{5}-1}{2}+1$$

Simplify. The value x=\frac{-\sqrt{5}-1}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\left(\frac{1}{2}-\frac{1}{2}\times 5^{\frac{1}{2}}\right)=-\frac{1}{2}\times 5^{\frac{1}{2}}+\frac{1}{2}$$

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

$$\sqrt{\frac{\sqrt{5}-1}{2}+2}=\frac{\sqrt{5}-1}{2}+1$$

Simplify. The value x=\frac{\sqrt{5}-1}{2} satisfies the equation.

$$\frac{1}{2}+\frac{1}{2}\times 5^{\frac{1}{2}}=\frac{1}{2}\times 5^{\frac{1}{2}}+\frac{1}{2}$$

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

$$x=\frac{\sqrt{5}-1}{2}$$