Solve 9-22x+12x^2

$$9-22x+12 { x }^{ 2 }$$

Factor

$$12\left(x-\frac{11-\sqrt{13}}{12}\right)\left(x-\frac{\sqrt{13}+11}{12}\right)$$

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.

$$12x^{2}-22x+9=0$$

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.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 12\times 9}}{2\times 12}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 12\times 9}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-22\right)±\sqrt{484-48\times 9}}{2\times 12}$$

Multiply -48 times 9.

$$x=\frac{-\left(-22\right)±\sqrt{484-432}}{2\times 12}$$

Add 484 to -432.

$$x=\frac{-\left(-22\right)±\sqrt{52}}{2\times 12}$$

Take the square root of 52.

$$x=\frac{-\left(-22\right)±2\sqrt{13}}{2\times 12}$$

The opposite of -22 is 22.

$$x=\frac{22±2\sqrt{13}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{22±2\sqrt{13}}{24}$$

Now solve the equation x=\frac{22±2\sqrt{13}}{24} when ± is plus. Add 22 to 2\sqrt{13}.

$$x=\frac{2\sqrt{13}+22}{24}$$

Divide 22+2\sqrt{13} by 24.

$$x=\frac{\sqrt{13}+11}{12}$$

Now solve the equation x=\frac{22±2\sqrt{13}}{24} when ± is minus. Subtract 2\sqrt{13} from 22.

$$x=\frac{22-2\sqrt{13}}{24}$$

Divide 22-2\sqrt{13} by 24.

$$x=\frac{11-\sqrt{13}}{12}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11+\sqrt{13}}{12} for x_{1} and \frac{11-\sqrt{13}}{12} for x_{2}.

$$12x^{2}-22x+9=12\left(x-\frac{\sqrt{13}+11}{12}\right)\left(x-\frac{11-\sqrt{13}}{12}\right)$$

Evaluate

$$12x^{2}-22x+9$$