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

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

Solve for x

$$x=3\sqrt{209}+34\approx 77.370496884$$

$$x=34-3\sqrt{209}\approx -9.370496884$$

Steps Using the Quadratic Formula

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.

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

Subtract 350 from both sides of the equation.

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

Subtracting 350 from itself leaves 0.

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

Subtract 350 from 1800.

$$-2x^{2}+136x+1450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 136 for b, and 1450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-136±\sqrt{136^{2}-4\left(-2\right)\times 1450}}{2\left(-2\right)}$$

Square 136.

$$x=\frac{-136±\sqrt{18496-4\left(-2\right)\times 1450}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-136±\sqrt{18496+8\times 1450}}{2\left(-2\right)}$$

Multiply 8 times 1450.

$$x=\frac{-136±\sqrt{18496+11600}}{2\left(-2\right)}$$

Add 18496 to 11600.

$$x=\frac{-136±\sqrt{30096}}{2\left(-2\right)}$$

Take the square root of 30096.

$$x=\frac{-136±12\sqrt{209}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-136±12\sqrt{209}}{-4}$$

Now solve the equation x=\frac{-136±12\sqrt{209}}{-4} when ± is plus. Add -136 to 12\sqrt{209}.

$$x=\frac{12\sqrt{209}-136}{-4}$$

Divide -136+12\sqrt{209} by -4.

$$x=34-3\sqrt{209}$$

Now solve the equation x=\frac{-136±12\sqrt{209}}{-4} when ± is minus. Subtract 12\sqrt{209} from -136.

$$x=\frac{-12\sqrt{209}-136}{-4}$$

Divide -136-12\sqrt{209} by -4.

$$x=3\sqrt{209}+34$$

The equation is now solved.

$$x=34-3\sqrt{209} x=3\sqrt{209}+34$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Subtract 1800 from both sides of the equation.

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

Subtracting 1800 from itself leaves 0.

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

Subtract 1800 from 350.

$$-2x^{2}+136x=-1450$$

Divide both sides by -2.

$$\frac{-2x^{2}+136x}{-2}=-\frac{1450}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{136}{-2}x=-\frac{1450}{-2}$$

Divide 136 by -2.

$$x^{2}-68x=-\frac{1450}{-2}$$

Divide -1450 by -2.

$$x^{2}-68x=725$$

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.

$$x^{2}-68x+\left(-34\right)^{2}=725+\left(-34\right)^{2}$$

Square -34.

$$x^{2}-68x+1156=725+1156$$

Add 725 to 1156.

$$x^{2}-68x+1156=1881$$

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}.

$$\left(x-34\right)^{2}=1881$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-34\right)^{2}}=\sqrt{1881}$$

Simplify.

$$x-34=3\sqrt{209} x-34=-3\sqrt{209}$$

Add 34 to both sides of the equation.

$$x=3\sqrt{209}+34 x=34-3\sqrt{209}$$