Solve x^2-15x=-150

$$x ^ { 2 } - 15 x = - 150$$

Solve for x (complex solution)

$$x=\frac{15+5\sqrt{15}i}{2}\approx 7.5+9.682458366i$$

$$x=\frac{-5\sqrt{15}i+15}{2}\approx 7.5-9.682458366i$$

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.

$$x^{2}-15x=-150$$

Add 150 to both sides of the equation.

$$x^{2}-15x-\left(-150\right)=-150-\left(-150\right)$$

Subtracting -150 from itself leaves 0.

$$x^{2}-15x-\left(-150\right)=0$$

Subtract -150 from 0.

$$x^{2}-15x+150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -15 for b, and 150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 150}}{2}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 150}}{2}$$

Multiply -4 times 150.

$$x=\frac{-\left(-15\right)±\sqrt{225-600}}{2}$$

Add 225 to -600.

$$x=\frac{-\left(-15\right)±\sqrt{-375}}{2}$$

Take the square root of -375.

$$x=\frac{-\left(-15\right)±5\sqrt{15}i}{2}$$

The opposite of -15 is 15.

$$x=\frac{15±5\sqrt{15}i}{2}$$

Now solve the equation x=\frac{15±5\sqrt{15}i}{2} when ± is plus. Add 15 to 5i\sqrt{15}.

$$x=\frac{15+5\sqrt{15}i}{2}$$

Now solve the equation x=\frac{15±5\sqrt{15}i}{2} when ± is minus. Subtract 5i\sqrt{15} from 15.

$$x=\frac{-5\sqrt{15}i+15}{2}$$

The equation is now solved.

$$x=\frac{15+5\sqrt{15}i}{2} x=\frac{-5\sqrt{15}i+15}{2}$$

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.

$$x^{2}-15x=-150$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-150+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-150+\frac{225}{4}$$

Add -150 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=-\frac{375}{4}$$

Factor x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=-\frac{375}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{-\frac{375}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{5\sqrt{15}i}{2} x-\frac{15}{2}=-\frac{5\sqrt{15}i}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=\frac{15+5\sqrt{15}i}{2} x=\frac{-5\sqrt{15}i+15}{2}$$