Solve u^2-88*5u+1242*977=0

$$u ^ { 2 } - 88 \cdot 5 u + 1242 \cdot 977 = 0$$

Solve for u

$$u=220+\sqrt{1165034}i\approx 220+1079.367407327i$$

$$u=-\sqrt{1165034}i+220\approx 220-1079.367407327i$$

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.

$$u^{2}-440u+1213434=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -440 for b, and 1213434 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{-\left(-440\right)±\sqrt{\left(-440\right)^{2}-4\times 1213434}}{2}$$

Square -440.

$$u=\frac{-\left(-440\right)±\sqrt{193600-4\times 1213434}}{2}$$

Multiply -4 times 1213434.

$$u=\frac{-\left(-440\right)±\sqrt{193600-4853736}}{2}$$

Add 193600 to -4853736.

$$u=\frac{-\left(-440\right)±\sqrt{-4660136}}{2}$$

Take the square root of -4660136.

$$u=\frac{-\left(-440\right)±2\sqrt{1165034}i}{2}$$

The opposite of -440 is 440.

$$u=\frac{440±2\sqrt{1165034}i}{2}$$

Now solve the equation u=\frac{440±2\sqrt{1165034}i}{2} when ± is plus. Add 440 to 2i\sqrt{1165034}.

$$u=\frac{440+2\sqrt{1165034}i}{2}$$

Divide 440+2i\sqrt{1165034} by 2.

$$u=220+\sqrt{1165034}i$$

Now solve the equation u=\frac{440±2\sqrt{1165034}i}{2} when ± is minus. Subtract 2i\sqrt{1165034} from 440.

$$u=\frac{-2\sqrt{1165034}i+440}{2}$$

Divide 440-2i\sqrt{1165034} by 2.

$$u=-\sqrt{1165034}i+220$$

The equation is now solved.

$$u=220+\sqrt{1165034}i u=-\sqrt{1165034}i+220$$

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.

$$u^{2}-440u+1213434=0$$

Subtract 1213434 from both sides of the equation.

$$u^{2}-440u+1213434-1213434=-1213434$$

Subtracting 1213434 from itself leaves 0.

$$u^{2}-440u=-1213434$$

Divide -440, the coefficient of the x term, by 2 to get -220. Then add the square of -220 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$u^{2}-440u+\left(-220\right)^{2}=-1213434+\left(-220\right)^{2}$$

Square -220.

$$u^{2}-440u+48400=-1213434+48400$$

Add -1213434 to 48400.

$$u^{2}-440u+48400=-1165034$$

Factor u^{2}-440u+48400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(u-220\right)^{2}=-1165034$$

Take the square root of both sides of the equation.

$$\sqrt{\left(u-220\right)^{2}}=\sqrt{-1165034}$$

Simplify.

$$u-220=\sqrt{1165034}i u-220=-\sqrt{1165034}i$$

Add 220 to both sides of the equation.

$$u=220+\sqrt{1165034}i u=-\sqrt{1165034}i+220$$