Solve sqrt{x-5}-x=13

$$\sqrt{ x-5 } -x = 13$$

Solve for x (complex solution)

$$x=\frac{-25+\sqrt{71}i}{2}\approx -12.5+4.213074887i$$

Solution Steps

Subtract -x from both sides of the equation.

$$\sqrt{x-5}=13+x$$

Square both sides of the equation.

$$\left(\sqrt{x-5}\right)^{2}=\left(13+x\right)^{2}$$

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

$$x-5=\left(13+x\right)^{2}$$

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

$$x-5=169+26x+x^{2}$$

Subtract 169 from both sides.

$$x-5-169=26x+x^{2}$$

Subtract 169 from -5 to get -174.

$$x-174=26x+x^{2}$$

Subtract 26x from both sides.

$$x-174-26x=x^{2}$$

Combine x and -26x to get -25x.

$$-25x-174=x^{2}$$

Subtract x^{2} from both sides.

$$-25x-174-x^{2}=0$$

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.

$$-x^{2}-25x-174=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -25 for b, and -174 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\left(-1\right)\left(-174\right)}}{2\left(-1\right)}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\left(-1\right)\left(-174\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-25\right)±\sqrt{625+4\left(-174\right)}}{2\left(-1\right)}$$

Multiply 4 times -174.

$$x=\frac{-\left(-25\right)±\sqrt{625-696}}{2\left(-1\right)}$$

Add 625 to -696.

$$x=\frac{-\left(-25\right)±\sqrt{-71}}{2\left(-1\right)}$$

Take the square root of -71.

$$x=\frac{-\left(-25\right)±\sqrt{71}i}{2\left(-1\right)}$$

The opposite of -25 is 25.

$$x=\frac{25±\sqrt{71}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{25±\sqrt{71}i}{-2}$$

Now solve the equation x=\frac{25±\sqrt{71}i}{-2} when ± is plus. Add 25 to i\sqrt{71}.

$$x=\frac{25+\sqrt{71}i}{-2}$$

Divide 25+i\sqrt{71} by -2.

$$x=\frac{-\sqrt{71}i-25}{2}$$

Now solve the equation x=\frac{25±\sqrt{71}i}{-2} when ± is minus. Subtract i\sqrt{71} from 25.

$$x=\frac{-\sqrt{71}i+25}{-2}$$

Divide 25-i\sqrt{71} by -2.

$$x=\frac{-25+\sqrt{71}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{71}i-25}{2} x=\frac{-25+\sqrt{71}i}{2}$$

Substitute \frac{-\sqrt{71}i-25}{2} for x in the equation \sqrt{x-5}-x=13.

$$\sqrt{\frac{-\sqrt{71}i-25}{2}-5}-\frac{-\sqrt{71}i-25}{2}=13$$

Simplify. The value x=\frac{-\sqrt{71}i-25}{2} does not satisfy the equation.

$$12+i\times 71^{\frac{1}{2}}=13$$

Substitute \frac{-25+\sqrt{71}i}{2} for x in the equation \sqrt{x-5}-x=13.

$$\sqrt{\frac{-25+\sqrt{71}i}{2}-5}-\frac{-25+\sqrt{71}i}{2}=13$$

Simplify. The value x=\frac{-25+\sqrt{71}i}{2} satisfies the equation.

$$13=13$$

Equation \sqrt{x-5}=x+13 has a unique solution.

$$x=\frac{-25+\sqrt{71}i}{2}$$