Solve sqrt{5x+20}=8-x

$$\sqrt { 5 x + 20 } = 8 - x$$

Solve for x

$$x = \frac{21 - \sqrt{265}}{2} \approx 2.360589702$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{5x+20}\right)^{2}=\left(8-x\right)^{2}$$

Calculate \sqrt{5x+20} to the power of 2 and get 5x+20.

$$5x+20=\left(8-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-x\right)^{2}.

$$5x+20=64-16x+x^{2}$$

Subtract 64 from both sides.

$$5x+20-64=-16x+x^{2}$$

Subtract 64 from 20 to get -44.

$$5x-44=-16x+x^{2}$$

Add 16x to both sides.

$$5x-44+16x=x^{2}$$

Combine 5x and 16x to get 21x.

$$21x-44=x^{2}$$

Subtract x^{2} from both sides.

$$21x-44-x^{2}=0$$

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.

$$-x^{2}+21x-44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 21 for b, and -44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\left(-1\right)\left(-44\right)}}{2\left(-1\right)}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\left(-1\right)\left(-44\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-21±\sqrt{441+4\left(-44\right)}}{2\left(-1\right)}$$

Multiply 4 times -44.

$$x=\frac{-21±\sqrt{441-176}}{2\left(-1\right)}$$

Add 441 to -176.

$$x=\frac{-21±\sqrt{265}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-21±\sqrt{265}}{-2}$$

Now solve the equation x=\frac{-21±\sqrt{265}}{-2} when ± is plus. Add -21 to \sqrt{265}.

$$x=\frac{\sqrt{265}-21}{-2}$$

Divide -21+\sqrt{265} by -2.

$$x=\frac{21-\sqrt{265}}{2}$$

Now solve the equation x=\frac{-21±\sqrt{265}}{-2} when ± is minus. Subtract \sqrt{265} from -21.

$$x=\frac{-\sqrt{265}-21}{-2}$$

Divide -21-\sqrt{265} by -2.

$$x=\frac{\sqrt{265}+21}{2}$$

The equation is now solved.

$$x=\frac{21-\sqrt{265}}{2} x=\frac{\sqrt{265}+21}{2}$$

Substitute \frac{21-\sqrt{265}}{2} for x in the equation \sqrt{5x+20}=8-x.

$$\sqrt{5\times \frac{21-\sqrt{265}}{2}+20}=8-\frac{21-\sqrt{265}}{2}$$

Simplify. The value x=\frac{21-\sqrt{265}}{2} satisfies the equation.

$$-\left(\frac{5}{2}-\frac{1}{2}\times 265^{\frac{1}{2}}\right)=-\frac{5}{2}+\frac{1}{2}\times 265^{\frac{1}{2}}$$

Substitute \frac{\sqrt{265}+21}{2} for x in the equation \sqrt{5x+20}=8-x.

$$\sqrt{5\times \frac{\sqrt{265}+21}{2}+20}=8-\frac{\sqrt{265}+21}{2}$$

Simplify. The value x=\frac{\sqrt{265}+21}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{5}{2}+\frac{1}{2}\times 265^{\frac{1}{2}}=-\frac{5}{2}-\frac{1}{2}\times 265^{\frac{1}{2}}$$

Equation \sqrt{5x+20}=8-x has a unique solution.

$$x=\frac{21-\sqrt{265}}{2}$$