Solve sqrt{x+2}=sqrt{x^2-3}

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

Solve for x

$$x=\frac{1-\sqrt{21}}{2}\approx -1.791287847$$

$$x = \frac{\sqrt{21} + 1}{2} \approx 2.791287847$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Subtract x^{2} from both sides.

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

Add 3 to both sides.

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

Add 2 and 3 to get 5.

$$x+5-x^{2}=0$$

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.

$$-x^{2}+x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1±\sqrt{1+4\times 5}}{2\left(-1\right)}$$

Multiply 4 times 5.

$$x=\frac{-1±\sqrt{1+20}}{2\left(-1\right)}$$

Add 1 to 20.

$$x=\frac{-1±\sqrt{21}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1±\sqrt{21}}{-2}$$

Now solve the equation x=\frac{-1±\sqrt{21}}{-2} when ± is plus. Add -1 to \sqrt{21}.

$$x=\frac{\sqrt{21}-1}{-2}$$

Divide -1+\sqrt{21} by -2.

$$x=\frac{1-\sqrt{21}}{2}$$

Now solve the equation x=\frac{-1±\sqrt{21}}{-2} when ± is minus. Subtract \sqrt{21} from -1.

$$x=\frac{-\sqrt{21}-1}{-2}$$

Divide -1-\sqrt{21} by -2.

$$x=\frac{\sqrt{21}+1}{2}$$

The equation is now solved.

$$x=\frac{1-\sqrt{21}}{2} x=\frac{\sqrt{21}+1}{2}$$

Substitute \frac{1-\sqrt{21}}{2} for x in the equation \sqrt{x+2}=\sqrt{x^{2}-3}.

$$\sqrt{\frac{1-\sqrt{21}}{2}+2}=\sqrt{\left(\frac{1-\sqrt{21}}{2}\right)^{2}-3}$$

Simplify. The value x=\frac{1-\sqrt{21}}{2} satisfies the equation.

$$\frac{1}{2}\times 7^{\frac{1}{2}}-\frac{1}{2}\times 3^{\frac{1}{2}}=\frac{1}{2}\times 7^{\frac{1}{2}}-\frac{1}{2}\times 3^{\frac{1}{2}}$$

Substitute \frac{\sqrt{21}+1}{2} for x in the equation \sqrt{x+2}=\sqrt{x^{2}-3}.

$$\sqrt{\frac{\sqrt{21}+1}{2}+2}=\sqrt{\left(\frac{\sqrt{21}+1}{2}\right)^{2}-3}$$

Simplify. The value x=\frac{\sqrt{21}+1}{2} satisfies the equation.

$$\frac{1}{2}\times 7^{\frac{1}{2}}+\frac{1}{2}\times 3^{\frac{1}{2}}=\frac{1}{2}\times 7^{\frac{1}{2}}+\frac{1}{2}\times 3^{\frac{1}{2}}$$

List all solutions of \sqrt{x+2}=\sqrt{x^{2}-3}.

$$x=\frac{1-\sqrt{21}}{2} x=\frac{\sqrt{21}+1}{2}$$