Solve sqrt{x}+2x=2

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

Solve for x

$$x=\frac{9-\sqrt{17}}{8}\approx 0.609611797$$

Solution Steps

Subtract 2x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

$$x=4-8x+4x^{2}$$

Subtract 4 from both sides.

$$x-4=-8x+4x^{2}$$

Add 8x to both sides.

$$x-4+8x=4x^{2}$$

Combine x and 8x to get 9x.

$$9x-4=4x^{2}$$

Subtract 4x^{2} from both sides.

$$9x-4-4x^{2}=0$$

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.

$$-4x^{2}+9x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 9 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-9±\sqrt{81+16\left(-4\right)}}{2\left(-4\right)}$$

Multiply 16 times -4.

$$x=\frac{-9±\sqrt{81-64}}{2\left(-4\right)}$$

Add 81 to -64.

$$x=\frac{-9±\sqrt{17}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-9±\sqrt{17}}{-8}$$

Now solve the equation x=\frac{-9±\sqrt{17}}{-8} when ± is plus. Add -9 to \sqrt{17}.

$$x=\frac{\sqrt{17}-9}{-8}$$

Divide -9+\sqrt{17} by -8.

$$x=\frac{9-\sqrt{17}}{8}$$

Now solve the equation x=\frac{-9±\sqrt{17}}{-8} when ± is minus. Subtract \sqrt{17} from -9.

$$x=\frac{-\sqrt{17}-9}{-8}$$

Divide -9-\sqrt{17} by -8.

$$x=\frac{\sqrt{17}+9}{8}$$

The equation is now solved.

$$x=\frac{9-\sqrt{17}}{8} x=\frac{\sqrt{17}+9}{8}$$

Substitute \frac{9-\sqrt{17}}{8} for x in the equation \sqrt{x}+2x=2.

$$\sqrt{\frac{9-\sqrt{17}}{8}}+2\times \frac{9-\sqrt{17}}{8}=2$$

Simplify. The value x=\frac{9-\sqrt{17}}{8} satisfies the equation.

$$2=2$$

Substitute \frac{\sqrt{17}+9}{8} for x in the equation \sqrt{x}+2x=2.

$$\sqrt{\frac{\sqrt{17}+9}{8}}+2\times \frac{\sqrt{17}+9}{8}=2$$

Simplify. The value x=\frac{\sqrt{17}+9}{8} does not satisfy the equation.

$$\frac{5}{2}+\frac{1}{2}\times 17^{\frac{1}{2}}=2$$

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

$$x=\frac{9-\sqrt{17}}{8}$$