Solve (t^2-6t+2)+(5t^2-t-8)

$$( t ^ { 2 } - 6 t + 2 ) + ( 5 t ^ { 2 } - t - 8 )$$

Evaluate

$$6t^{2}-7t-6$$

Solution Steps

Combine t^{2} and 5t^{2} to get 6t^{2}.

$$6t^{2}-6t+2-t-8$$

Combine -6t and -t to get -7t.

$$6t^{2}-7t+2-8$$

Subtract 8 from 2 to get -6.

$$6t^{2}-7t-6$$

Factor

$$6\left(t-\frac{7-\sqrt{193}}{12}\right)\left(t-\frac{\sqrt{193}+7}{12}\right)$$

Steps Using the Quadratic Formula

Combine t^{2} and 5t^{2} to get 6t^{2}.

$$factor(6t^{2}-6t+2-t-8)$$

Combine -6t and -t to get -7t.

$$factor(6t^{2}-7t+2-8)$$

Subtract 8 from 2 to get -6.

$$factor(6t^{2}-7t-6)$$

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.

$$6t^{2}-7t-6=0$$

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.

$$t=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\left(-6\right)}}{2\times 6}$$

Square -7.

$$t=\frac{-\left(-7\right)±\sqrt{49-4\times 6\left(-6\right)}}{2\times 6}$$

Multiply -4 times 6.

$$t=\frac{-\left(-7\right)±\sqrt{49-24\left(-6\right)}}{2\times 6}$$

Multiply -24 times -6.

$$t=\frac{-\left(-7\right)±\sqrt{49+144}}{2\times 6}$$

Add 49 to 144.

$$t=\frac{-\left(-7\right)±\sqrt{193}}{2\times 6}$$

The opposite of -7 is 7.

$$t=\frac{7±\sqrt{193}}{2\times 6}$$

Multiply 2 times 6.

$$t=\frac{7±\sqrt{193}}{12}$$

Now solve the equation t=\frac{7±\sqrt{193}}{12} when ± is plus. Add 7 to \sqrt{193}.

$$t=\frac{\sqrt{193}+7}{12}$$

Now solve the equation t=\frac{7±\sqrt{193}}{12} when ± is minus. Subtract \sqrt{193} from 7.

$$t=\frac{7-\sqrt{193}}{12}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{193}}{12} for x_{1} and \frac{7-\sqrt{193}}{12} for x_{2}.

$$6t^{2}-7t-6=6\left(t-\frac{\sqrt{193}+7}{12}\right)\left(t-\frac{7-\sqrt{193}}{12}\right)$$