Solve 2x^2-150x+7500=0

$$2 { x }^{ 2 } -150x+7500=0$$

Solve for x (complex solution)

$$x=\frac{75+25\sqrt{15}i}{2}\approx 37.5+48.412291828i$$

$$x=\frac{-25\sqrt{15}i+75}{2}\approx 37.5-48.412291828i$$

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.

$$2x^{2}-150x+7500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -150 for b, and 7500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-150\right)±\sqrt{\left(-150\right)^{2}-4\times 2\times 7500}}{2\times 2}$$

Square -150.

$$x=\frac{-\left(-150\right)±\sqrt{22500-4\times 2\times 7500}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-150\right)±\sqrt{22500-8\times 7500}}{2\times 2}$$

Multiply -8 times 7500.

$$x=\frac{-\left(-150\right)±\sqrt{22500-60000}}{2\times 2}$$

Add 22500 to -60000.

$$x=\frac{-\left(-150\right)±\sqrt{-37500}}{2\times 2}$$

Take the square root of -37500.

$$x=\frac{-\left(-150\right)±50\sqrt{15}i}{2\times 2}$$

The opposite of -150 is 150.

$$x=\frac{150±50\sqrt{15}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{150±50\sqrt{15}i}{4}$$

Now solve the equation x=\frac{150±50\sqrt{15}i}{4} when ± is plus. Add 150 to 50i\sqrt{15}.

$$x=\frac{150+50\sqrt{15}i}{4}$$

Divide 150+50i\sqrt{15} by 4.

$$x=\frac{75+25\sqrt{15}i}{2}$$

Now solve the equation x=\frac{150±50\sqrt{15}i}{4} when ± is minus. Subtract 50i\sqrt{15} from 150.

$$x=\frac{-50\sqrt{15}i+150}{4}$$

Divide 150-50i\sqrt{15} by 4.

$$x=\frac{-25\sqrt{15}i+75}{2}$$

The equation is now solved.

$$x=\frac{75+25\sqrt{15}i}{2} x=\frac{-25\sqrt{15}i+75}{2}$$

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.

$$2x^{2}-150x+7500=0$$

Subtract 7500 from both sides of the equation.

$$2x^{2}-150x+7500-7500=-7500$$

Subtracting 7500 from itself leaves 0.

$$2x^{2}-150x=-7500$$

Divide both sides by 2.

$$\frac{2x^{2}-150x}{2}=-\frac{7500}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{150}{2}\right)x=-\frac{7500}{2}$$

Divide -150 by 2.

$$x^{2}-75x=-\frac{7500}{2}$$

Divide -7500 by 2.

$$x^{2}-75x=-3750$$

Divide -75, the coefficient of the x term, by 2 to get -\frac{75}{2}. Then add the square of -\frac{75}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-75x+\left(-\frac{75}{2}\right)^{2}=-3750+\left(-\frac{75}{2}\right)^{2}$$

Square -\frac{75}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-75x+\frac{5625}{4}=-3750+\frac{5625}{4}$$

Add -3750 to \frac{5625}{4}.

$$x^{2}-75x+\frac{5625}{4}=-\frac{9375}{4}$$

Factor x^{2}-75x+\frac{5625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{75}{2}\right)^{2}=-\frac{9375}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{75}{2}\right)^{2}}=\sqrt{-\frac{9375}{4}}$$

Simplify.

$$x-\frac{75}{2}=\frac{25\sqrt{15}i}{2} x-\frac{75}{2}=-\frac{25\sqrt{15}i}{2}$$

Add \frac{75}{2} to both sides of the equation.

$$x=\frac{75+25\sqrt{15}i}{2} x=\frac{-25\sqrt{15}i+75}{2}$$