Solve x+sqrt{x}=90

$$x+ \sqrt{ x } =90$$

Solve for x

$$x=81$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{x}=90-x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(90-x\right)^{2}$$

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

$$x=\left(90-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(90-x\right)^{2}.

$$x=8100-180x+x^{2}$$

Subtract 8100 from both sides.

$$x-8100=-180x+x^{2}$$

Add 180x to both sides.

$$x-8100+180x=x^{2}$$

Combine x and 180x to get 181x.

$$181x-8100=x^{2}$$

Subtract x^{2} from both sides.

$$181x-8100-x^{2}=0$$

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.

$$-x^{2}+181x-8100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 181 for b, and -8100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-181±\sqrt{181^{2}-4\left(-1\right)\left(-8100\right)}}{2\left(-1\right)}$$

Square 181.

$$x=\frac{-181±\sqrt{32761-4\left(-1\right)\left(-8100\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-181±\sqrt{32761+4\left(-8100\right)}}{2\left(-1\right)}$$

Multiply 4 times -8100.

$$x=\frac{-181±\sqrt{32761-32400}}{2\left(-1\right)}$$

Add 32761 to -32400.

$$x=\frac{-181±\sqrt{361}}{2\left(-1\right)}$$

Take the square root of 361.

$$x=\frac{-181±19}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-181±19}{-2}$$

Now solve the equation x=\frac{-181±19}{-2} when ± is plus. Add -181 to 19.

$$x=-\frac{162}{-2}$$

Divide -162 by -2.

$$x=81$$

Now solve the equation x=\frac{-181±19}{-2} when ± is minus. Subtract 19 from -181.

$$x=-\frac{200}{-2}$$

Divide -200 by -2.

$$x=100$$

The equation is now solved.

$$x=81 x=100$$

Substitute 81 for x in the equation x+\sqrt{x}=90.

$$81+\sqrt{81}=90$$

Simplify. The value x=81 satisfies the equation.

$$90=90$$

Substitute 100 for x in the equation x+\sqrt{x}=90.

$$100+\sqrt{100}=90$$

Simplify. The value x=100 does not satisfy the equation.

$$110=90$$

Equation \sqrt{x}=90-x has a unique solution.

$$x=81$$