Solve x^2-115x+5046=0

$${ x }^{ 2 } -115x+5046=0$$

Solve for x (complex solution)

$$x=\frac{115+\sqrt{6959}i}{2}\approx 57.5+41.710310476i$$

$$x=\frac{-\sqrt{6959}i+115}{2}\approx 57.5-41.710310476i$$

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.

$$x^{2}-115x+5046=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -115 for b, and 5046 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-115\right)±\sqrt{\left(-115\right)^{2}-4\times 5046}}{2}$$

Square -115.

$$x=\frac{-\left(-115\right)±\sqrt{13225-4\times 5046}}{2}$$

Multiply -4 times 5046.

$$x=\frac{-\left(-115\right)±\sqrt{13225-20184}}{2}$$

Add 13225 to -20184.

$$x=\frac{-\left(-115\right)±\sqrt{-6959}}{2}$$

Take the square root of -6959.

$$x=\frac{-\left(-115\right)±\sqrt{6959}i}{2}$$

The opposite of -115 is 115.

$$x=\frac{115±\sqrt{6959}i}{2}$$

Now solve the equation x=\frac{115±\sqrt{6959}i}{2} when ± is plus. Add 115 to i\sqrt{6959}.

$$x=\frac{115+\sqrt{6959}i}{2}$$

Now solve the equation x=\frac{115±\sqrt{6959}i}{2} when ± is minus. Subtract i\sqrt{6959} from 115.

$$x=\frac{-\sqrt{6959}i+115}{2}$$

The equation is now solved.

$$x=\frac{115+\sqrt{6959}i}{2} x=\frac{-\sqrt{6959}i+115}{2}$$

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.

$$x^{2}-115x+5046=0$$

Subtract 5046 from both sides of the equation.

$$x^{2}-115x+5046-5046=-5046$$

Subtracting 5046 from itself leaves 0.

$$x^{2}-115x=-5046$$

Divide -115, the coefficient of the x term, by 2 to get -\frac{115}{2}. Then add the square of -\frac{115}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-115x+\left(-\frac{115}{2}\right)^{2}=-5046+\left(-\frac{115}{2}\right)^{2}$$

Square -\frac{115}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-115x+\frac{13225}{4}=-5046+\frac{13225}{4}$$

Add -5046 to \frac{13225}{4}.

$$x^{2}-115x+\frac{13225}{4}=-\frac{6959}{4}$$

Factor x^{2}-115x+\frac{13225}{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}.

$$\left(x-\frac{115}{2}\right)^{2}=-\frac{6959}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{115}{2}\right)^{2}}=\sqrt{-\frac{6959}{4}}$$

Simplify.

$$x-\frac{115}{2}=\frac{\sqrt{6959}i}{2} x-\frac{115}{2}=-\frac{\sqrt{6959}i}{2}$$

Add \frac{115}{2} to both sides of the equation.

$$x=\frac{115+\sqrt{6959}i}{2} x=\frac{-\sqrt{6959}i+115}{2}$$