Solve sqrt{x}+x=5

$$\sqrt{ x } +x=5$$

Solve for x

$$x = \frac{11 - \sqrt{21}}{2} \approx 3.208712153$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{x}=5-x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(5-x\right)^{2}$$

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

$$x=\left(5-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$x=25-10x+x^{2}$$

Subtract 25 from both sides.

$$x-25=-10x+x^{2}$$

Add 10x to both sides.

$$x-25+10x=x^{2}$$

Combine x and 10x to get 11x.

$$11x-25=x^{2}$$

Subtract x^{2} from both sides.

$$11x-25-x^{2}=0$$

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.

$$-x^{2}+11x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 11 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-1\right)\left(-25\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-11±\sqrt{121+4\left(-25\right)}}{2\left(-1\right)}$$

Multiply 4 times -25.

$$x=\frac{-11±\sqrt{121-100}}{2\left(-1\right)}$$

Add 121 to -100.

$$x=\frac{-11±\sqrt{21}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-11±\sqrt{21}}{-2}$$

Now solve the equation x=\frac{-11±\sqrt{21}}{-2} when ± is plus. Add -11 to \sqrt{21}.

$$x=\frac{\sqrt{21}-11}{-2}$$

Divide -11+\sqrt{21} by -2.

$$x=\frac{11-\sqrt{21}}{2}$$

Now solve the equation x=\frac{-11±\sqrt{21}}{-2} when ± is minus. Subtract \sqrt{21} from -11.

$$x=\frac{-\sqrt{21}-11}{-2}$$

Divide -11-\sqrt{21} by -2.

$$x=\frac{\sqrt{21}+11}{2}$$

The equation is now solved.

$$x=\frac{11-\sqrt{21}}{2} x=\frac{\sqrt{21}+11}{2}$$

Substitute \frac{11-\sqrt{21}}{2} for x in the equation \sqrt{x}+x=5.

$$\sqrt{\frac{11-\sqrt{21}}{2}}+\frac{11-\sqrt{21}}{2}=5$$

Simplify. The value x=\frac{11-\sqrt{21}}{2} satisfies the equation.

$$5=5$$

Substitute \frac{\sqrt{21}+11}{2} for x in the equation \sqrt{x}+x=5.

$$\sqrt{\frac{\sqrt{21}+11}{2}}+\frac{\sqrt{21}+11}{2}=5$$

Simplify. The value x=\frac{\sqrt{21}+11}{2} does not satisfy the equation.

$$6+21^{\frac{1}{2}}=5$$

Equation \sqrt{x}=5-x has a unique solution.

$$x=\frac{11-\sqrt{21}}{2}$$