Solve x+sqrt{x}=110

$$x+ \sqrt{ x } =110$$

Solve for x

$$x=100$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{x}=110-x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(110-x\right)^{2}$$

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

$$x=\left(110-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(110-x\right)^{2}.

$$x=12100-220x+x^{2}$$

Subtract 12100 from both sides.

$$x-12100=-220x+x^{2}$$

Add 220x to both sides.

$$x-12100+220x=x^{2}$$

Combine x and 220x to get 221x.

$$221x-12100=x^{2}$$

Subtract x^{2} from both sides.

$$221x-12100-x^{2}=0$$

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.

$$-x^{2}+221x-12100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 221 for b, and -12100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-221±\sqrt{221^{2}-4\left(-1\right)\left(-12100\right)}}{2\left(-1\right)}$$

Square 221.

$$x=\frac{-221±\sqrt{48841-4\left(-1\right)\left(-12100\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-221±\sqrt{48841+4\left(-12100\right)}}{2\left(-1\right)}$$

Multiply 4 times -12100.

$$x=\frac{-221±\sqrt{48841-48400}}{2\left(-1\right)}$$

Add 48841 to -48400.

$$x=\frac{-221±\sqrt{441}}{2\left(-1\right)}$$

Take the square root of 441.

$$x=\frac{-221±21}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-221±21}{-2}$$

Now solve the equation x=\frac{-221±21}{-2} when ± is plus. Add -221 to 21.

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

Divide -200 by -2.

$$x=100$$

Now solve the equation x=\frac{-221±21}{-2} when ± is minus. Subtract 21 from -221.

$$x=-\frac{242}{-2}$$

Divide -242 by -2.

$$x=121$$

The equation is now solved.

$$x=100 x=121$$

Substitute 100 for x in the equation x+\sqrt{x}=110.

$$100+\sqrt{100}=110$$

Simplify. The value x=100 satisfies the equation.

$$110=110$$

Substitute 121 for x in the equation x+\sqrt{x}=110.

$$121+\sqrt{121}=110$$

Simplify. The value x=121 does not satisfy the equation.

$$132=110$$

Equation \sqrt{x}=110-x has a unique solution.

$$x=100$$