Solve sqrt{58+x}=x+5

$$\sqrt{ 58+x } =x+5$$

Solve for x

$$x = \frac{\sqrt{213} - 9}{2} \approx 2.79725976$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{58+x}\right)^{2}=\left(x+5\right)^{2}$$

Calculate \sqrt{58+x} to the power of 2 and get 58+x.

$$58+x=\left(x+5\right)^{2}$$

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

$$58+x=x^{2}+10x+25$$

Subtract x^{2} from both sides.

$$58+x-x^{2}=10x+25$$

Subtract 10x from both sides.

$$58+x-x^{2}-10x=25$$

Combine x and -10x to get -9x.

$$58-9x-x^{2}=25$$

Subtract 25 from both sides.

$$58-9x-x^{2}-25=0$$

Subtract 25 from 58 to get 33.

$$33-9x-x^{2}=0$$

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.

$$-x^{2}-9x+33=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -9 for b, and 33 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-1\right)\times 33}}{2\left(-1\right)}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-1\right)\times 33}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-9\right)±\sqrt{81+4\times 33}}{2\left(-1\right)}$$

Multiply 4 times 33.

$$x=\frac{-\left(-9\right)±\sqrt{81+132}}{2\left(-1\right)}$$

Add 81 to 132.

$$x=\frac{-\left(-9\right)±\sqrt{213}}{2\left(-1\right)}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{213}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{9±\sqrt{213}}{-2}$$

Now solve the equation x=\frac{9±\sqrt{213}}{-2} when ± is plus. Add 9 to \sqrt{213}.

$$x=\frac{\sqrt{213}+9}{-2}$$

Divide 9+\sqrt{213} by -2.

$$x=\frac{-\sqrt{213}-9}{2}$$

Now solve the equation x=\frac{9±\sqrt{213}}{-2} when ± is minus. Subtract \sqrt{213} from 9.

$$x=\frac{9-\sqrt{213}}{-2}$$

Divide 9-\sqrt{213} by -2.

$$x=\frac{\sqrt{213}-9}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{213}-9}{2} x=\frac{\sqrt{213}-9}{2}$$

Substitute \frac{-\sqrt{213}-9}{2} for x in the equation \sqrt{58+x}=x+5.

$$\sqrt{58+\frac{-\sqrt{213}-9}{2}}=\frac{-\sqrt{213}-9}{2}+5$$

Simplify. The value x=\frac{-\sqrt{213}-9}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\left(\frac{1}{2}-\frac{1}{2}\times 213^{\frac{1}{2}}\right)=-\frac{1}{2}\times 213^{\frac{1}{2}}+\frac{1}{2}$$

Substitute \frac{\sqrt{213}-9}{2} for x in the equation \sqrt{58+x}=x+5.

$$\sqrt{58+\frac{\sqrt{213}-9}{2}}=\frac{\sqrt{213}-9}{2}+5$$

Simplify. The value x=\frac{\sqrt{213}-9}{2} satisfies the equation.

$$\frac{1}{2}+\frac{1}{2}\times 213^{\frac{1}{2}}=\frac{1}{2}\times 213^{\frac{1}{2}}+\frac{1}{2}$$

Equation \sqrt{x+58}=x+5 has a unique solution.

$$x=\frac{\sqrt{213}-9}{2}$$