Solve x+sqrt{x}-13=0

$$x+ \sqrt{ x } -13=0$$

Solve for x

$$x = \frac{27 - \sqrt{53}}{2} \approx 9.859945055$$

Solution Steps

Subtract x-13 from both sides of the equation.

$$\sqrt{x}=-\left(x-13\right)$$

To find the opposite of x-13, find the opposite of each term.

$$\sqrt{x}=-x-\left(-13\right)$$

The opposite of -13 is 13.

$$\sqrt{x}=-x+13$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(-x+13\right)^{2}$$

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

$$x=\left(-x+13\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-x+13\right)^{2}.

$$x=x^{2}-26x+169$$

Subtract x^{2} from both sides.

$$x-x^{2}=-26x+169$$

Add 26x to both sides.

$$x-x^{2}+26x=169$$

Combine x and 26x to get 27x.

$$27x-x^{2}=169$$

Subtract 169 from both sides.

$$27x-x^{2}-169=0$$

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.

$$-x^{2}+27x-169=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 27 for b, and -169 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27±\sqrt{27^{2}-4\left(-1\right)\left(-169\right)}}{2\left(-1\right)}$$

Square 27.

$$x=\frac{-27±\sqrt{729-4\left(-1\right)\left(-169\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-27±\sqrt{729+4\left(-169\right)}}{2\left(-1\right)}$$

Multiply 4 times -169.

$$x=\frac{-27±\sqrt{729-676}}{2\left(-1\right)}$$

Add 729 to -676.

$$x=\frac{-27±\sqrt{53}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-27±\sqrt{53}}{-2}$$

Now solve the equation x=\frac{-27±\sqrt{53}}{-2} when ± is plus. Add -27 to \sqrt{53}.

$$x=\frac{\sqrt{53}-27}{-2}$$

Divide -27+\sqrt{53} by -2.

$$x=\frac{27-\sqrt{53}}{2}$$

Now solve the equation x=\frac{-27±\sqrt{53}}{-2} when ± is minus. Subtract \sqrt{53} from -27.

$$x=\frac{-\sqrt{53}-27}{-2}$$

Divide -27-\sqrt{53} by -2.

$$x=\frac{\sqrt{53}+27}{2}$$

The equation is now solved.

$$x=\frac{27-\sqrt{53}}{2} x=\frac{\sqrt{53}+27}{2}$$

Substitute \frac{27-\sqrt{53}}{2} for x in the equation x+\sqrt{x}-13=0.

$$\frac{27-\sqrt{53}}{2}+\sqrt{\frac{27-\sqrt{53}}{2}}-13=0$$

Simplify. The value x=\frac{27-\sqrt{53}}{2} satisfies the equation.

$$0=0$$

Substitute \frac{\sqrt{53}+27}{2} for x in the equation x+\sqrt{x}-13=0.

$$\frac{\sqrt{53}+27}{2}+\sqrt{\frac{\sqrt{53}+27}{2}}-13=0$$

Simplify. The value x=\frac{\sqrt{53}+27}{2} does not satisfy the equation.

$$53^{\frac{1}{2}}+1=0$$

Equation \sqrt{x}=13-x has a unique solution.

$$x=\frac{27-\sqrt{53}}{2}$$