Solve 4+6t^2-13t

$$4 + 6 t ^ { 2 } - 13 t$$

Factor

$$6\left(t-\frac{13-\sqrt{73}}{12}\right)\left(t-\frac{\sqrt{73}+13}{12}\right)$$

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.

$$6t^{2}-13t+4=0$$

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.

$$t=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 6\times 4}}{2\times 6}$$

Square -13.

$$t=\frac{-\left(-13\right)±\sqrt{169-4\times 6\times 4}}{2\times 6}$$

Multiply -4 times 6.

$$t=\frac{-\left(-13\right)±\sqrt{169-24\times 4}}{2\times 6}$$

Multiply -24 times 4.

$$t=\frac{-\left(-13\right)±\sqrt{169-96}}{2\times 6}$$

Add 169 to -96.

$$t=\frac{-\left(-13\right)±\sqrt{73}}{2\times 6}$$

The opposite of -13 is 13.

$$t=\frac{13±\sqrt{73}}{2\times 6}$$

Multiply 2 times 6.

$$t=\frac{13±\sqrt{73}}{12}$$

Now solve the equation t=\frac{13±\sqrt{73}}{12} when ± is plus. Add 13 to \sqrt{73}.

$$t=\frac{\sqrt{73}+13}{12}$$

Now solve the equation t=\frac{13±\sqrt{73}}{12} when ± is minus. Subtract \sqrt{73} from 13.

$$t=\frac{13-\sqrt{73}}{12}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13+\sqrt{73}}{12} for x_{1} and \frac{13-\sqrt{73}}{12} for x_{2}.

$$6t^{2}-13t+4=6\left(t-\frac{\sqrt{73}+13}{12}\right)\left(t-\frac{13-\sqrt{73}}{12}\right)$$

Evaluate

$$6t^{2}-13t+4$$