Solve sqrt{5-x}=x+17

$$\sqrt { 5 - x } = x + 17$$

Solve for x

$$x=\frac{\sqrt{89}-35}{2}\approx -12.783009434$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{5-x}\right)^{2}=\left(x+17\right)^{2}$$

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

$$5-x=\left(x+17\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+17\right)^{2}.

$$5-x=x^{2}+34x+289$$

Subtract x^{2} from both sides.

$$5-x-x^{2}=34x+289$$

Subtract 34x from both sides.

$$5-x-x^{2}-34x=289$$

Combine -x and -34x to get -35x.

$$5-35x-x^{2}=289$$

Subtract 289 from both sides.

$$5-35x-x^{2}-289=0$$

Subtract 289 from 5 to get -284.

$$-284-35x-x^{2}=0$$

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.

$$-x^{2}-35x-284=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -35 for b, and -284 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-35\right)±\sqrt{\left(-35\right)^{2}-4\left(-1\right)\left(-284\right)}}{2\left(-1\right)}$$

Square -35.

$$x=\frac{-\left(-35\right)±\sqrt{1225-4\left(-1\right)\left(-284\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-35\right)±\sqrt{1225+4\left(-284\right)}}{2\left(-1\right)}$$

Multiply 4 times -284.

$$x=\frac{-\left(-35\right)±\sqrt{1225-1136}}{2\left(-1\right)}$$

Add 1225 to -1136.

$$x=\frac{-\left(-35\right)±\sqrt{89}}{2\left(-1\right)}$$

The opposite of -35 is 35.

$$x=\frac{35±\sqrt{89}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{35±\sqrt{89}}{-2}$$

Now solve the equation x=\frac{35±\sqrt{89}}{-2} when ± is plus. Add 35 to \sqrt{89}.

$$x=\frac{\sqrt{89}+35}{-2}$$

Divide 35+\sqrt{89} by -2.

$$x=\frac{-\sqrt{89}-35}{2}$$

Now solve the equation x=\frac{35±\sqrt{89}}{-2} when ± is minus. Subtract \sqrt{89} from 35.

$$x=\frac{35-\sqrt{89}}{-2}$$

Divide 35-\sqrt{89} by -2.

$$x=\frac{\sqrt{89}-35}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{89}-35}{2} x=\frac{\sqrt{89}-35}{2}$$

Substitute \frac{-\sqrt{89}-35}{2} for x in the equation \sqrt{5-x}=x+17.

$$\sqrt{5-\frac{-\sqrt{89}-35}{2}}=\frac{-\sqrt{89}-35}{2}+17$$

Simplify. The value x=\frac{-\sqrt{89}-35}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{1}{2}+\frac{1}{2}\times 89^{\frac{1}{2}}=-\frac{1}{2}\times 89^{\frac{1}{2}}-\frac{1}{2}$$

Substitute \frac{\sqrt{89}-35}{2} for x in the equation \sqrt{5-x}=x+17.

$$\sqrt{5-\frac{\sqrt{89}-35}{2}}=\frac{\sqrt{89}-35}{2}+17$$

Simplify. The value x=\frac{\sqrt{89}-35}{2} satisfies the equation.

$$-\left(\frac{1}{2}-\frac{1}{2}\times 89^{\frac{1}{2}}\right)=\frac{1}{2}\times 89^{\frac{1}{2}}-\frac{1}{2}$$

Equation \sqrt{5-x}=x+17 has a unique solution.

$$x=\frac{\sqrt{89}-35}{2}$$