Solve 12x^2-40x+14

$$12 { x }^{ 2 } -40x+14$$

Factor

$$12\left(x-\left(-\frac{\sqrt{58}}{6}+\frac{5}{3}\right)\right)\left(x-\left(\frac{\sqrt{58}}{6}+\frac{5}{3}\right)\right)$$

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.

$$12x^{2}-40x+14=0$$

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.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 12\times 14}}{2\times 12}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 12\times 14}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-\left(-40\right)±\sqrt{1600-48\times 14}}{2\times 12}$$

Multiply -48 times 14.

$$x=\frac{-\left(-40\right)±\sqrt{1600-672}}{2\times 12}$$

Add 1600 to -672.

$$x=\frac{-\left(-40\right)±\sqrt{928}}{2\times 12}$$

Take the square root of 928.

$$x=\frac{-\left(-40\right)±4\sqrt{58}}{2\times 12}$$

The opposite of -40 is 40.

$$x=\frac{40±4\sqrt{58}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{40±4\sqrt{58}}{24}$$

Now solve the equation x=\frac{40±4\sqrt{58}}{24} when ± is plus. Add 40 to 4\sqrt{58}.

$$x=\frac{4\sqrt{58}+40}{24}$$

Divide 40+4\sqrt{58} by 24.

$$x=\frac{\sqrt{58}}{6}+\frac{5}{3}$$

Now solve the equation x=\frac{40±4\sqrt{58}}{24} when ± is minus. Subtract 4\sqrt{58} from 40.

$$x=\frac{40-4\sqrt{58}}{24}$$

Divide 40-4\sqrt{58} by 24.

$$x=-\frac{\sqrt{58}}{6}+\frac{5}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{3}+\frac{\sqrt{58}}{6} for x_{1} and \frac{5}{3}-\frac{\sqrt{58}}{6} for x_{2}.

$$12x^{2}-40x+14=12\left(x-\left(\frac{\sqrt{58}}{6}+\frac{5}{3}\right)\right)\left(x-\left(-\frac{\sqrt{58}}{6}+\frac{5}{3}\right)\right)$$

Evaluate

$$12x^{2}-40x+14$$