Solve left(-xright)^2-14x+240=0

$${ \left(-x \right) }^{ 2 } -14x+240 = 0$$

Solve for x (complex solution)

$$x=7+\sqrt{191}i\approx 7+13.820274961i$$

$$x=-\sqrt{191}i+7\approx 7-13.820274961i$$

Steps Using the Quadratic Formula

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}-14x+240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and 240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 240}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 240}}{2}$$

Multiply -4 times 240.

$$x=\frac{-\left(-14\right)±\sqrt{196-960}}{2}$$

Add 196 to -960.

$$x=\frac{-\left(-14\right)±\sqrt{-764}}{2}$$

Take the square root of -764.

$$x=\frac{-\left(-14\right)±2\sqrt{191}i}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{191}i}{2}$$

Now solve the equation x=\frac{14±2\sqrt{191}i}{2} when ± is plus. Add 14 to 2i\sqrt{191}.

$$x=\frac{14+2\sqrt{191}i}{2}$$

Divide 14+2i\sqrt{191} by 2.

$$x=7+\sqrt{191}i$$

Now solve the equation x=\frac{14±2\sqrt{191}i}{2} when ± is minus. Subtract 2i\sqrt{191} from 14.

$$x=\frac{-2\sqrt{191}i+14}{2}$$

Divide 14-2i\sqrt{191} by 2.

$$x=-\sqrt{191}i+7$$

The equation is now solved.

$$x=7+\sqrt{191}i x=-\sqrt{191}i+7$$

Steps for Completing the Square

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}-14x+240=0$$

Subtract 240 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-14x=-240$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-240+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-240+49$$

Add -240 to 49.

$$x^{2}-14x+49=-191$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=-191$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{-191}$$

Simplify.

$$x-7=\sqrt{191}i x-7=-\sqrt{191}i$$

Add 7 to both sides of the equation.

$$x=7+\sqrt{191}i x=-\sqrt{191}i+7$$