Solve -2x^2+136x-1800=3500000

$$-2 { x }^{ 2 } +136x-1800 = 3500000$$

Solve for x (complex solution)

$$x=34+12\sqrt{12151}i\approx 34+1322.778893088i$$

$$x=-12\sqrt{12151}i+34\approx 34-1322.778893088i$$

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.

$$-2x^{2}+136x-1800=3500000$$

Subtract 3500000 from both sides of the equation.

$$-2x^{2}+136x-1800-3500000=3500000-3500000$$

Subtracting 3500000 from itself leaves 0.

$$-2x^{2}+136x-1800-3500000=0$$

Subtract 3500000 from -1800.

$$-2x^{2}+136x-3501800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 136 for b, and -3501800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-136±\sqrt{136^{2}-4\left(-2\right)\left(-3501800\right)}}{2\left(-2\right)}$$

Square 136.

$$x=\frac{-136±\sqrt{18496-4\left(-2\right)\left(-3501800\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-136±\sqrt{18496+8\left(-3501800\right)}}{2\left(-2\right)}$$

Multiply 8 times -3501800.

$$x=\frac{-136±\sqrt{18496-28014400}}{2\left(-2\right)}$$

Add 18496 to -28014400.

$$x=\frac{-136±\sqrt{-27995904}}{2\left(-2\right)}$$

Take the square root of -27995904.

$$x=\frac{-136±48\sqrt{12151}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-136±48\sqrt{12151}i}{-4}$$

Now solve the equation x=\frac{-136±48\sqrt{12151}i}{-4} when ± is plus. Add -136 to 48i\sqrt{12151}.

$$x=\frac{-136+48\sqrt{12151}i}{-4}$$

Divide -136+48i\sqrt{12151} by -4.

$$x=-12\sqrt{12151}i+34$$

Now solve the equation x=\frac{-136±48\sqrt{12151}i}{-4} when ± is minus. Subtract 48i\sqrt{12151} from -136.

$$x=\frac{-48\sqrt{12151}i-136}{-4}$$

Divide -136-48i\sqrt{12151} by -4.

$$x=34+12\sqrt{12151}i$$

The equation is now solved.

$$x=-12\sqrt{12151}i+34 x=34+12\sqrt{12151}i$$

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.

$$-2x^{2}+136x-1800=3500000$$

Add 1800 to both sides of the equation.

$$-2x^{2}+136x-1800-\left(-1800\right)=3500000-\left(-1800\right)$$

Subtracting -1800 from itself leaves 0.

$$-2x^{2}+136x=3500000-\left(-1800\right)$$

Subtract -1800 from 3500000.

$$-2x^{2}+136x=3501800$$

Divide both sides by -2.

$$\frac{-2x^{2}+136x}{-2}=\frac{3501800}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{136}{-2}x=\frac{3501800}{-2}$$

Divide 136 by -2.

$$x^{2}-68x=\frac{3501800}{-2}$$

Divide 3501800 by -2.

$$x^{2}-68x=-1750900$$

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.

$$x^{2}-68x+\left(-34\right)^{2}=-1750900+\left(-34\right)^{2}$$

Square -34.

$$x^{2}-68x+1156=-1750900+1156$$

Add -1750900 to 1156.

$$x^{2}-68x+1156=-1749744$$

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}.

$$\left(x-34\right)^{2}=-1749744$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-34\right)^{2}}=\sqrt{-1749744}$$

Simplify.

$$x-34=12\sqrt{12151}i x-34=-12\sqrt{12151}i$$

Add 34 to both sides of the equation.

$$x=34+12\sqrt{12151}i x=-12\sqrt{12151}i+34$$