Solve -2x^2+136x-1800-350=0

$$-2 { x }^{ 2 } +136x-1800-350=0$$

Solve for x

$$x=25$$

$$x=43$$

Steps Using the Quadratic Formula

Subtract 350 from -1800 to get -2150.

$$-2x^{2}+136x-2150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 136 for b, and -2150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-136±\sqrt{136^{2}-4\left(-2\right)\left(-2150\right)}}{2\left(-2\right)}$$

Square 136.

$$x=\frac{-136±\sqrt{18496-4\left(-2\right)\left(-2150\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-136±\sqrt{18496+8\left(-2150\right)}}{2\left(-2\right)}$$

Multiply 8 times -2150.

$$x=\frac{-136±\sqrt{18496-17200}}{2\left(-2\right)}$$

Add 18496 to -17200.

$$x=\frac{-136±\sqrt{1296}}{2\left(-2\right)}$$

Take the square root of 1296.

$$x=\frac{-136±36}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-136±36}{-4}$$

Now solve the equation x=\frac{-136±36}{-4} when ± is plus. Add -136 to 36.

$$x=-\frac{100}{-4}$$

Divide -100 by -4.

$$x=25$$

Now solve the equation x=\frac{-136±36}{-4} when ± is minus. Subtract 36 from -136.

$$x=-\frac{172}{-4}$$

Divide -172 by -4.

$$x=43$$

The equation is now solved.

$$x=25 x=43$$

Steps for Completing the Square

Subtract 350 from -1800 to get -2150.

$$-2x^{2}+136x-2150=0$$

Add 2150 to both sides. Anything plus zero gives itself.

$$-2x^{2}+136x=2150$$

Divide both sides by -2.

$$\frac{-2x^{2}+136x}{-2}=\frac{2150}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{136}{-2}x=\frac{2150}{-2}$$

Divide 136 by -2.

$$x^{2}-68x=\frac{2150}{-2}$$

Divide 2150 by -2.

$$x^{2}-68x=-1075$$

Divide -68, the coefficient of the x term, by 2 to get -34. Then add the square of -34 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-68x+\left(-34\right)^{2}=-1075+\left(-34\right)^{2}$$

Square -34.

$$x^{2}-68x+1156=-1075+1156$$

Add -1075 to 1156.

$$x^{2}-68x+1156=81$$

Factor x^{2}-68x+1156. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-34\right)^{2}=81$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-34\right)^{2}}=\sqrt{81}$$

Simplify.

$$x-34=9 x-34=-9$$

Add 34 to both sides of the equation.

$$x=43 x=25$$