Solve 11x^2-54x-192

$$11 { x }^{ 2 } -54x-192$$

Factor

$$11\left(x-\frac{27-\sqrt{2841}}{11}\right)\left(x-\frac{\sqrt{2841}+27}{11}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$11x^{2}-54x-192=0$$

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.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 11\left(-192\right)}}{2\times 11}$$

Square -54.

$$x=\frac{-\left(-54\right)±\sqrt{2916-4\times 11\left(-192\right)}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-54\right)±\sqrt{2916-44\left(-192\right)}}{2\times 11}$$

Multiply -44 times -192.

$$x=\frac{-\left(-54\right)±\sqrt{2916+8448}}{2\times 11}$$

Add 2916 to 8448.

$$x=\frac{-\left(-54\right)±\sqrt{11364}}{2\times 11}$$

Take the square root of 11364.

$$x=\frac{-\left(-54\right)±2\sqrt{2841}}{2\times 11}$$

The opposite of -54 is 54.

$$x=\frac{54±2\sqrt{2841}}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{54±2\sqrt{2841}}{22}$$

Now solve the equation x=\frac{54±2\sqrt{2841}}{22} when ± is plus. Add 54 to 2\sqrt{2841}.

$$x=\frac{2\sqrt{2841}+54}{22}$$

Divide 54+2\sqrt{2841} by 22.

$$x=\frac{\sqrt{2841}+27}{11}$$

Now solve the equation x=\frac{54±2\sqrt{2841}}{22} when ± is minus. Subtract 2\sqrt{2841} from 54.

$$x=\frac{54-2\sqrt{2841}}{22}$$

Divide 54-2\sqrt{2841} by 22.

$$x=\frac{27-\sqrt{2841}}{11}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{27+\sqrt{2841}}{11} for x_{1} and \frac{27-\sqrt{2841}}{11} for x_{2}.

$$11x^{2}-54x-192=11\left(x-\frac{\sqrt{2841}+27}{11}\right)\left(x-\frac{27-\sqrt{2841}}{11}\right)$$

Evaluate

$$11x^{2}-54x-192$$