Solve -x^2+18x+4=x^2-2x+8

$$- x ^ { 2 } + 18 x + 4 = x ^ { 2 } - 2 x + 8$$

Solve for x

$$x=\sqrt{23}+5\approx 9.795831523$$

$$x=5-\sqrt{23}\approx 0.204168477$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$-x^{2}+18x+4-x^{2}=-2x+8$$

Add 2x to both sides.

$$-x^{2}+18x+4-x^{2}+2x=8$$

Combine 18x and 2x to get 20x.

$$-x^{2}+20x+4-x^{2}=8$$

Subtract 8 from both sides.

$$-x^{2}+20x+4-x^{2}-8=0$$

Subtract 8 from 4 to get -4.

$$-x^{2}+20x-4-x^{2}=0$$

Combine -x^{2} and -x^{2} to get -2x^{2}.

$$-2x^{2}+20x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 20 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-2\right)\left(-4\right)}}{2\left(-2\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-2\right)\left(-4\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-20±\sqrt{400+8\left(-4\right)}}{2\left(-2\right)}$$

Multiply 8 times -4.

$$x=\frac{-20±\sqrt{400-32}}{2\left(-2\right)}$$

Add 400 to -32.

$$x=\frac{-20±\sqrt{368}}{2\left(-2\right)}$$

Take the square root of 368.

$$x=\frac{-20±4\sqrt{23}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-20±4\sqrt{23}}{-4}$$

Now solve the equation x=\frac{-20±4\sqrt{23}}{-4} when ± is plus. Add -20 to 4\sqrt{23}.

$$x=\frac{4\sqrt{23}-20}{-4}$$

Divide -20+4\sqrt{23} by -4.

$$x=5-\sqrt{23}$$

Now solve the equation x=\frac{-20±4\sqrt{23}}{-4} when ± is minus. Subtract 4\sqrt{23} from -20.

$$x=\frac{-4\sqrt{23}-20}{-4}$$

Divide -20-4\sqrt{23} by -4.

$$x=\sqrt{23}+5$$

The equation is now solved.

$$x=5-\sqrt{23} x=\sqrt{23}+5$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$-x^{2}+18x+4-x^{2}=-2x+8$$

Add 2x to both sides.

$$-x^{2}+18x+4-x^{2}+2x=8$$

Combine 18x and 2x to get 20x.

$$-x^{2}+20x+4-x^{2}=8$$

Subtract 4 from both sides.

$$-x^{2}+20x-x^{2}=8-4$$

Subtract 4 from 8 to get 4.

$$-x^{2}+20x-x^{2}=4$$

Combine -x^{2} and -x^{2} to get -2x^{2}.

$$-2x^{2}+20x=4$$

Divide both sides by -2.

$$\frac{-2x^{2}+20x}{-2}=\frac{4}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{20}{-2}x=\frac{4}{-2}$$

Divide 20 by -2.

$$x^{2}-10x=\frac{4}{-2}$$

Divide 4 by -2.

$$x^{2}-10x=-2$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-2+\left(-5\right)^{2}$$

Square -5.

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

Add -2 to 25.

$$x^{2}-10x+25=23$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=23$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{23}$$

Simplify.

$$x-5=\sqrt{23} x-5=-\sqrt{23}$$

Add 5 to both sides of the equation.

$$x=\sqrt{23}+5 x=5-\sqrt{23}$$