Solve x-sqrt{x}=36

$$x- \sqrt{ x } =36$$

Solve for x

$$x = \frac{\sqrt{145} + 73}{2} \approx 42.520797289$$

Solution Steps

Subtract x from both sides of the equation.

$$-\sqrt{x}=36-x$$

Square both sides of the equation.

$$\left(-\sqrt{x}\right)^{2}=\left(36-x\right)^{2}$$

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

$$\left(-1\right)^{2}\left(\sqrt{x}\right)^{2}=\left(36-x\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{x}\right)^{2}=\left(36-x\right)^{2}$$

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

$$1x=\left(36-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(36-x\right)^{2}.

$$1x=1296-72x+x^{2}$$

Reorder the terms.

$$x=x^{2}-72x+1296$$

Subtract x^{2} from both sides.

$$x-x^{2}=-72x+1296$$

Add 72x to both sides.

$$x-x^{2}+72x=1296$$

Combine x and 72x to get 73x.

$$73x-x^{2}=1296$$

Subtract 1296 from both sides.

$$73x-x^{2}-1296=0$$

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.

$$-x^{2}+73x-1296=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 73 for b, and -1296 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-73±\sqrt{73^{2}-4\left(-1\right)\left(-1296\right)}}{2\left(-1\right)}$$

Square 73.

$$x=\frac{-73±\sqrt{5329-4\left(-1\right)\left(-1296\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-73±\sqrt{5329+4\left(-1296\right)}}{2\left(-1\right)}$$

Multiply 4 times -1296.

$$x=\frac{-73±\sqrt{5329-5184}}{2\left(-1\right)}$$

Add 5329 to -5184.

$$x=\frac{-73±\sqrt{145}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-73±\sqrt{145}}{-2}$$

Now solve the equation x=\frac{-73±\sqrt{145}}{-2} when ± is plus. Add -73 to \sqrt{145}.

$$x=\frac{\sqrt{145}-73}{-2}$$

Divide -73+\sqrt{145} by -2.

$$x=\frac{73-\sqrt{145}}{2}$$

Now solve the equation x=\frac{-73±\sqrt{145}}{-2} when ± is minus. Subtract \sqrt{145} from -73.

$$x=\frac{-\sqrt{145}-73}{-2}$$

Divide -73-\sqrt{145} by -2.

$$x=\frac{\sqrt{145}+73}{2}$$

The equation is now solved.

$$x=\frac{73-\sqrt{145}}{2} x=\frac{\sqrt{145}+73}{2}$$

Substitute \frac{73-\sqrt{145}}{2} for x in the equation x-\sqrt{x}=36.

$$\frac{73-\sqrt{145}}{2}-\sqrt{\frac{73-\sqrt{145}}{2}}=36$$

Simplify. The value x=\frac{73-\sqrt{145}}{2} does not satisfy the equation.

$$37-145^{\frac{1}{2}}=36$$

Substitute \frac{\sqrt{145}+73}{2} for x in the equation x-\sqrt{x}=36.

$$\frac{\sqrt{145}+73}{2}-\sqrt{\frac{\sqrt{145}+73}{2}}=36$$

Simplify. The value x=\frac{\sqrt{145}+73}{2} satisfies the equation.

$$36=36$$

Equation -\sqrt{x}=36-x has a unique solution.

$$x=\frac{\sqrt{145}+73}{2}$$