Solve sqrt{x^2+2x+10}=2x

$$\sqrt { x ^ { 2 } + 2 x + 10 } = 2 x$$

Solve for x

$$x = \frac{\sqrt{31} + 1}{3} \approx 2.189254788$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x^{2}+2x+10}\right)^{2}=\left(2x\right)^{2}$$

Calculate \sqrt{x^{2}+2x+10} to the power of 2 and get x^{2}+2x+10.

$$x^{2}+2x+10=\left(2x\right)^{2}$$

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

$$x^{2}+2x+10=2^{2}x^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}+2x+10=4x^{2}$$

Subtract 4x^{2} from both sides.

$$x^{2}+2x+10-4x^{2}=0$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}+2x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 2 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-3\right)\times 10}}{2\left(-3\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-3\right)\times 10}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-2±\sqrt{4+12\times 10}}{2\left(-3\right)}$$

Multiply 12 times 10.

$$x=\frac{-2±\sqrt{4+120}}{2\left(-3\right)}$$

Add 4 to 120.

$$x=\frac{-2±\sqrt{124}}{2\left(-3\right)}$$

Take the square root of 124.

$$x=\frac{-2±2\sqrt{31}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-2±2\sqrt{31}}{-6}$$

Now solve the equation x=\frac{-2±2\sqrt{31}}{-6} when ± is plus. Add -2 to 2\sqrt{31}.

$$x=\frac{2\sqrt{31}-2}{-6}$$

Divide -2+2\sqrt{31} by -6.

$$x=\frac{1-\sqrt{31}}{3}$$

Now solve the equation x=\frac{-2±2\sqrt{31}}{-6} when ± is minus. Subtract 2\sqrt{31} from -2.

$$x=\frac{-2\sqrt{31}-2}{-6}$$

Divide -2-2\sqrt{31} by -6.

$$x=\frac{\sqrt{31}+1}{3}$$

The equation is now solved.

$$x=\frac{1-\sqrt{31}}{3} x=\frac{\sqrt{31}+1}{3}$$

Substitute \frac{1-\sqrt{31}}{3} for x in the equation \sqrt{x^{2}+2x+10}=2x.

$$\sqrt{\left(\frac{1-\sqrt{31}}{3}\right)^{2}+2\times \frac{1-\sqrt{31}}{3}+10}=2\times \frac{1-\sqrt{31}}{3}$$

Simplify. The value x=\frac{1-\sqrt{31}}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{2}{3}\times 31^{\frac{1}{2}}-\frac{2}{3}=\frac{2}{3}-\frac{2}{3}\times 31^{\frac{1}{2}}$$

Substitute \frac{\sqrt{31}+1}{3} for x in the equation \sqrt{x^{2}+2x+10}=2x.

$$\sqrt{\left(\frac{\sqrt{31}+1}{3}\right)^{2}+2\times \frac{\sqrt{31}+1}{3}+10}=2\times \frac{\sqrt{31}+1}{3}$$

Simplify. The value x=\frac{\sqrt{31}+1}{3} satisfies the equation.

$$\frac{2}{3}\times 31^{\frac{1}{2}}+\frac{2}{3}=\frac{2}{3}\times 31^{\frac{1}{2}}+\frac{2}{3}$$

Equation \sqrt{x^{2}+2x+10}=2x has a unique solution.

$$x=\frac{\sqrt{31}+1}{3}$$