Solve 44x^2-240x+12096=0

$$44 { x }^{ 2 } -240x+12096=0$$

Solve for x (complex solution)

$$x=\frac{30+6\sqrt{899}i}{11}\approx 2.727272727+16.354542928i$$

$$x=\frac{-6\sqrt{899}i+30}{11}\approx 2.727272727-16.354542928i$$

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.

$$44x^{2}-240x+12096=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 44 for a, -240 for b, and 12096 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-240\right)±\sqrt{\left(-240\right)^{2}-4\times 44\times 12096}}{2\times 44}$$

Square -240.

$$x=\frac{-\left(-240\right)±\sqrt{57600-4\times 44\times 12096}}{2\times 44}$$

Multiply -4 times 44.

$$x=\frac{-\left(-240\right)±\sqrt{57600-176\times 12096}}{2\times 44}$$

Multiply -176 times 12096.

$$x=\frac{-\left(-240\right)±\sqrt{57600-2128896}}{2\times 44}$$

Add 57600 to -2128896.

$$x=\frac{-\left(-240\right)±\sqrt{-2071296}}{2\times 44}$$

Take the square root of -2071296.

$$x=\frac{-\left(-240\right)±48\sqrt{899}i}{2\times 44}$$

The opposite of -240 is 240.

$$x=\frac{240±48\sqrt{899}i}{2\times 44}$$

Multiply 2 times 44.

$$x=\frac{240±48\sqrt{899}i}{88}$$

Now solve the equation x=\frac{240±48\sqrt{899}i}{88} when ± is plus. Add 240 to 48i\sqrt{899}.

$$x=\frac{240+48\sqrt{899}i}{88}$$

Divide 240+48i\sqrt{899} by 88.

$$x=\frac{30+6\sqrt{899}i}{11}$$

Now solve the equation x=\frac{240±48\sqrt{899}i}{88} when ± is minus. Subtract 48i\sqrt{899} from 240.

$$x=\frac{-48\sqrt{899}i+240}{88}$$

Divide 240-48i\sqrt{899} by 88.

$$x=\frac{-6\sqrt{899}i+30}{11}$$

The equation is now solved.

$$x=\frac{30+6\sqrt{899}i}{11} x=\frac{-6\sqrt{899}i+30}{11}$$

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.

$$44x^{2}-240x+12096=0$$

Subtract 12096 from both sides of the equation.

$$44x^{2}-240x+12096-12096=-12096$$

Subtracting 12096 from itself leaves 0.

$$44x^{2}-240x=-12096$$

Divide both sides by 44.

$$\frac{44x^{2}-240x}{44}=-\frac{12096}{44}$$

Dividing by 44 undoes the multiplication by 44.

$$x^{2}+\left(-\frac{240}{44}\right)x=-\frac{12096}{44}$$

Reduce the fraction \frac{-240}{44} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{60}{11}x=-\frac{12096}{44}$$

Reduce the fraction \frac{-12096}{44} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{60}{11}x=-\frac{3024}{11}$$

Divide -\frac{60}{11}, the coefficient of the x term, by 2 to get -\frac{30}{11}. Then add the square of -\frac{30}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{60}{11}x+\left(-\frac{30}{11}\right)^{2}=-\frac{3024}{11}+\left(-\frac{30}{11}\right)^{2}$$

Square -\frac{30}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{60}{11}x+\frac{900}{121}=-\frac{3024}{11}+\frac{900}{121}$$

Add -\frac{3024}{11} to \frac{900}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{60}{11}x+\frac{900}{121}=-\frac{32364}{121}$$

Factor x^{2}-\frac{60}{11}x+\frac{900}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{30}{11}\right)^{2}=-\frac{32364}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{30}{11}\right)^{2}}=\sqrt{-\frac{32364}{121}}$$

Simplify.

$$x-\frac{30}{11}=\frac{6\sqrt{899}i}{11} x-\frac{30}{11}=-\frac{6\sqrt{899}i}{11}$$

Add \frac{30}{11} to both sides of the equation.

$$x=\frac{30+6\sqrt{899}i}{11} x=\frac{-6\sqrt{899}i+30}{11}$$