Solve 2x+sqrt{x}=526

$$2x+ \sqrt{ x } =526$$

Solve for x

$$x = \frac{2105 - \sqrt{4209}}{8} \approx 255.015399208$$

Solution Steps

Subtract 2x from both sides of the equation.

$$\sqrt{x}=526-2x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(526-2x\right)^{2}$$

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

$$x=\left(526-2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(526-2x\right)^{2}.

$$x=276676-2104x+4x^{2}$$

Subtract 276676 from both sides.

$$x-276676=-2104x+4x^{2}$$

Add 2104x to both sides.

$$x-276676+2104x=4x^{2}$$

Combine x and 2104x to get 2105x.

$$2105x-276676=4x^{2}$$

Subtract 4x^{2} from both sides.

$$2105x-276676-4x^{2}=0$$

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.

$$-4x^{2}+2105x-276676=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 2105 for b, and -276676 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2105±\sqrt{2105^{2}-4\left(-4\right)\left(-276676\right)}}{2\left(-4\right)}$$

Square 2105.

$$x=\frac{-2105±\sqrt{4431025-4\left(-4\right)\left(-276676\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-2105±\sqrt{4431025+16\left(-276676\right)}}{2\left(-4\right)}$$

Multiply 16 times -276676.

$$x=\frac{-2105±\sqrt{4431025-4426816}}{2\left(-4\right)}$$

Add 4431025 to -4426816.

$$x=\frac{-2105±\sqrt{4209}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-2105±\sqrt{4209}}{-8}$$

Now solve the equation x=\frac{-2105±\sqrt{4209}}{-8} when ± is plus. Add -2105 to \sqrt{4209}.

$$x=\frac{\sqrt{4209}-2105}{-8}$$

Divide -2105+\sqrt{4209} by -8.

$$x=\frac{2105-\sqrt{4209}}{8}$$

Now solve the equation x=\frac{-2105±\sqrt{4209}}{-8} when ± is minus. Subtract \sqrt{4209} from -2105.

$$x=\frac{-\sqrt{4209}-2105}{-8}$$

Divide -2105-\sqrt{4209} by -8.

$$x=\frac{\sqrt{4209}+2105}{8}$$

The equation is now solved.

$$x=\frac{2105-\sqrt{4209}}{8} x=\frac{\sqrt{4209}+2105}{8}$$

Substitute \frac{2105-\sqrt{4209}}{8} for x in the equation 2x+\sqrt{x}=526.

$$2\times \frac{2105-\sqrt{4209}}{8}+\sqrt{\frac{2105-\sqrt{4209}}{8}}=526$$

Simplify. The value x=\frac{2105-\sqrt{4209}}{8} satisfies the equation.

$$526=526$$

Substitute \frac{\sqrt{4209}+2105}{8} for x in the equation 2x+\sqrt{x}=526.

$$2\times \frac{\sqrt{4209}+2105}{8}+\sqrt{\frac{\sqrt{4209}+2105}{8}}=526$$

Simplify. The value x=\frac{\sqrt{4209}+2105}{8} does not satisfy the equation.

$$\frac{1}{2}\times 4209^{\frac{1}{2}}+\frac{1053}{2}=526$$

Equation \sqrt{x}=526-2x has a unique solution.

$$x=\frac{2105-\sqrt{4209}}{8}$$