Solve x^2-12x^2+48x-120=0

$$x ^ { 2 } - 12 x ^ { 2 } + 48 x - 120 = 0$$

Solve for x (complex solution)

$$x=\frac{-2\sqrt{186}i+24}{11}\approx 2.181818182-2.479669399i$$

$$x=\frac{24+2\sqrt{186}i}{11}\approx 2.181818182+2.479669399i$$

Steps Using the Quadratic Formula

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$-11x^{2}+48x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -11 for a, 48 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\left(-11\right)\left(-120\right)}}{2\left(-11\right)}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\left(-11\right)\left(-120\right)}}{2\left(-11\right)}$$

Multiply -4 times -11.

$$x=\frac{-48±\sqrt{2304+44\left(-120\right)}}{2\left(-11\right)}$$

Multiply 44 times -120.

$$x=\frac{-48±\sqrt{2304-5280}}{2\left(-11\right)}$$

Add 2304 to -5280.

$$x=\frac{-48±\sqrt{-2976}}{2\left(-11\right)}$$

Take the square root of -2976.

$$x=\frac{-48±4\sqrt{186}i}{2\left(-11\right)}$$

Multiply 2 times -11.

$$x=\frac{-48±4\sqrt{186}i}{-22}$$

Now solve the equation x=\frac{-48±4\sqrt{186}i}{-22} when ± is plus. Add -48 to 4i\sqrt{186}.

$$x=\frac{-48+4\sqrt{186}i}{-22}$$

Divide -48+4i\sqrt{186} by -22.

$$x=\frac{-2\sqrt{186}i+24}{11}$$

Now solve the equation x=\frac{-48±4\sqrt{186}i}{-22} when ± is minus. Subtract 4i\sqrt{186} from -48.

$$x=\frac{-4\sqrt{186}i-48}{-22}$$

Divide -48-4i\sqrt{186} by -22.

$$x=\frac{24+2\sqrt{186}i}{11}$$

The equation is now solved.

$$x=\frac{-2\sqrt{186}i+24}{11} x=\frac{24+2\sqrt{186}i}{11}$$

Steps for Completing the Square

Combine x^{2} and -12x^{2} to get -11x^{2}.

$$-11x^{2}+48x-120=0$$

Add 120 to both sides. Anything plus zero gives itself.

$$-11x^{2}+48x=120$$

Divide both sides by -11.

$$\frac{-11x^{2}+48x}{-11}=\frac{120}{-11}$$

Dividing by -11 undoes the multiplication by -11.

$$x^{2}+\frac{48}{-11}x=\frac{120}{-11}$$

Divide 48 by -11.

$$x^{2}-\frac{48}{11}x=\frac{120}{-11}$$

Divide 120 by -11.

$$x^{2}-\frac{48}{11}x=-\frac{120}{11}$$

Divide -\frac{48}{11}, the coefficient of the x term, by 2 to get -\frac{24}{11}. Then add the square of -\frac{24}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{48}{11}x+\left(-\frac{24}{11}\right)^{2}=-\frac{120}{11}+\left(-\frac{24}{11}\right)^{2}$$

Square -\frac{24}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{48}{11}x+\frac{576}{121}=-\frac{120}{11}+\frac{576}{121}$$

Add -\frac{120}{11} to \frac{576}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{48}{11}x+\frac{576}{121}=-\frac{744}{121}$$

Factor x^{2}-\frac{48}{11}x+\frac{576}{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}.

$$\left(x-\frac{24}{11}\right)^{2}=-\frac{744}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{24}{11}\right)^{2}}=\sqrt{-\frac{744}{121}}$$

Simplify.

$$x-\frac{24}{11}=\frac{2\sqrt{186}i}{11} x-\frac{24}{11}=-\frac{2\sqrt{186}i}{11}$$

Add \frac{24}{11} to both sides of the equation.

$$x=\frac{24+2\sqrt{186}i}{11} x=\frac{-2\sqrt{186}i+24}{11}$$