Solve (12x^2-13x-7)

$$( 12 x ^ { 2 } - 13 x - 7 )$$

Factor

$$12\left(x-\frac{13-\sqrt{505}}{24}\right)\left(x-\frac{\sqrt{505}+13}{24}\right)$$

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.

$$12x^{2}-13x-7=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 12\left(-7\right)}}{2\times 12}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 12\left(-7\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-13\right)±\sqrt{169-48\left(-7\right)}}{2\times 12}$$

Multiply -48 times -7.

$$x=\frac{-\left(-13\right)±\sqrt{169+336}}{2\times 12}$$

Add 169 to 336.

$$x=\frac{-\left(-13\right)±\sqrt{505}}{2\times 12}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{505}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{13±\sqrt{505}}{24}$$

Now solve the equation x=\frac{13±\sqrt{505}}{24} when ± is plus. Add 13 to \sqrt{505}.

$$x=\frac{\sqrt{505}+13}{24}$$

Now solve the equation x=\frac{13±\sqrt{505}}{24} when ± is minus. Subtract \sqrt{505} from 13.

$$x=\frac{13-\sqrt{505}}{24}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13+\sqrt{505}}{24} for x_{1} and \frac{13-\sqrt{505}}{24} for x_{2}.

$$12x^{2}-13x-7=12\left(x-\frac{\sqrt{505}+13}{24}\right)\left(x-\frac{13-\sqrt{505}}{24}\right)$$

Evaluate

$$12x^{2}-13x-7$$