Solve x+sqrt{x}=21

$$x + \sqrt { x } = 21$$

Solve for x

$$x = \frac{43 - \sqrt{85}}{2} \approx 16.890227771$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{x}=21-x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(21-x\right)^{2}$$

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

$$x=\left(21-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(21-x\right)^{2}.

$$x=441-42x+x^{2}$$

Subtract 441 from both sides.

$$x-441=-42x+x^{2}$$

Add 42x to both sides.

$$x-441+42x=x^{2}$$

Combine x and 42x to get 43x.

$$43x-441=x^{2}$$

Subtract x^{2} from both sides.

$$43x-441-x^{2}=0$$

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.

$$-x^{2}+43x-441=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 43 for b, and -441 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-43±\sqrt{43^{2}-4\left(-1\right)\left(-441\right)}}{2\left(-1\right)}$$

Square 43.

$$x=\frac{-43±\sqrt{1849-4\left(-1\right)\left(-441\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-43±\sqrt{1849+4\left(-441\right)}}{2\left(-1\right)}$$

Multiply 4 times -441.

$$x=\frac{-43±\sqrt{1849-1764}}{2\left(-1\right)}$$

Add 1849 to -1764.

$$x=\frac{-43±\sqrt{85}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-43±\sqrt{85}}{-2}$$

Now solve the equation x=\frac{-43±\sqrt{85}}{-2} when ± is plus. Add -43 to \sqrt{85}.

$$x=\frac{\sqrt{85}-43}{-2}$$

Divide -43+\sqrt{85} by -2.

$$x=\frac{43-\sqrt{85}}{2}$$

Now solve the equation x=\frac{-43±\sqrt{85}}{-2} when ± is minus. Subtract \sqrt{85} from -43.

$$x=\frac{-\sqrt{85}-43}{-2}$$

Divide -43-\sqrt{85} by -2.

$$x=\frac{\sqrt{85}+43}{2}$$

The equation is now solved.

$$x=\frac{43-\sqrt{85}}{2} x=\frac{\sqrt{85}+43}{2}$$

Substitute \frac{43-\sqrt{85}}{2} for x in the equation x+\sqrt{x}=21.

$$\frac{43-\sqrt{85}}{2}+\sqrt{\frac{43-\sqrt{85}}{2}}=21$$

Simplify. The value x=\frac{43-\sqrt{85}}{2} satisfies the equation.

$$21=21$$

Substitute \frac{\sqrt{85}+43}{2} for x in the equation x+\sqrt{x}=21.

$$\frac{\sqrt{85}+43}{2}+\sqrt{\frac{\sqrt{85}+43}{2}}=21$$

Simplify. The value x=\frac{\sqrt{85}+43}{2} does not satisfy the equation.

$$85^{\frac{1}{2}}+22=21$$

Equation \sqrt{x}=21-x has a unique solution.

$$x=\frac{43-\sqrt{85}}{2}$$