Solve sqrt{2x-8}+x=80

$$\sqrt { 2 x - 8 } + x = 80$$

Solve for x

$$x=81-3\sqrt{17}\approx 68.630683123$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{2x-8}=80-x$$

Square both sides of the equation.

$$\left(\sqrt{2x-8}\right)^{2}=\left(80-x\right)^{2}$$

Calculate \sqrt{2x-8} to the power of 2 and get 2x-8.

$$2x-8=\left(80-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(80-x\right)^{2}.

$$2x-8=6400-160x+x^{2}$$

Subtract 6400 from both sides.

$$2x-8-6400=-160x+x^{2}$$

Subtract 6400 from -8 to get -6408.

$$2x-6408=-160x+x^{2}$$

Add 160x to both sides.

$$2x-6408+160x=x^{2}$$

Combine 2x and 160x to get 162x.

$$162x-6408=x^{2}$$

Subtract x^{2} from both sides.

$$162x-6408-x^{2}=0$$

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.

$$-x^{2}+162x-6408=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 162 for b, and -6408 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-162±\sqrt{162^{2}-4\left(-1\right)\left(-6408\right)}}{2\left(-1\right)}$$

Square 162.

$$x=\frac{-162±\sqrt{26244-4\left(-1\right)\left(-6408\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-162±\sqrt{26244+4\left(-6408\right)}}{2\left(-1\right)}$$

Multiply 4 times -6408.

$$x=\frac{-162±\sqrt{26244-25632}}{2\left(-1\right)}$$

Add 26244 to -25632.

$$x=\frac{-162±\sqrt{612}}{2\left(-1\right)}$$

Take the square root of 612.

$$x=\frac{-162±6\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-162±6\sqrt{17}}{-2}$$

Now solve the equation x=\frac{-162±6\sqrt{17}}{-2} when ± is plus. Add -162 to 6\sqrt{17}.

$$x=\frac{6\sqrt{17}-162}{-2}$$

Divide -162+6\sqrt{17} by -2.

$$x=81-3\sqrt{17}$$

Now solve the equation x=\frac{-162±6\sqrt{17}}{-2} when ± is minus. Subtract 6\sqrt{17} from -162.

$$x=\frac{-6\sqrt{17}-162}{-2}$$

Divide -162-6\sqrt{17} by -2.

$$x=3\sqrt{17}+81$$

The equation is now solved.

$$x=81-3\sqrt{17} x=3\sqrt{17}+81$$

Substitute 81-3\sqrt{17} for x in the equation \sqrt{2x-8}+x=80.

$$\sqrt{2\left(81-3\sqrt{17}\right)-8}+81-3\sqrt{17}=80$$

Simplify. The value x=81-3\sqrt{17} satisfies the equation.

$$80=80$$

Substitute 3\sqrt{17}+81 for x in the equation \sqrt{2x-8}+x=80.

$$\sqrt{2\left(3\sqrt{17}+81\right)-8}+3\sqrt{17}+81=80$$

Simplify. The value x=3\sqrt{17}+81 does not satisfy the equation.

$$6\times 17^{\frac{1}{2}}+82=80$$

Equation \sqrt{2x-8}=80-x has a unique solution.

$$x=81-3\sqrt{17}$$