Solve (t^2-2t+5)+(-7t^2+4t+6)

$$( t ^ { 2 } - 2 t + 5 ) + ( - 7 t ^ { 2 } + 4 t + 6 )$$

Evaluate

$$11+2t-6t^{2}$$

Solution Steps

Combine t^{2} and -7t^{2} to get -6t^{2}.

$$-6t^{2}-2t+5+4t+6$$

Combine -2t and 4t to get 2t.

$$-6t^{2}+2t+5+6$$

Add 5 and 6 to get 11.

$$-6t^{2}+2t+11$$

Factor

$$-6\left(t-\frac{1-\sqrt{67}}{6}\right)\left(t-\frac{\sqrt{67}+1}{6}\right)$$

Steps Using the Quadratic Formula

Combine t^{2} and -7t^{2} to get -6t^{2}.

$$factor(-6t^{2}-2t+5+4t+6)$$

Combine -2t and 4t to get 2t.

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

Add 5 and 6 to get 11.

$$factor(-6t^{2}+2t+11)$$

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.

$$-6t^{2}+2t+11=0$$

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.

$$t=\frac{-2±\sqrt{2^{2}-4\left(-6\right)\times 11}}{2\left(-6\right)}$$

Square 2.

$$t=\frac{-2±\sqrt{4-4\left(-6\right)\times 11}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$t=\frac{-2±\sqrt{4+24\times 11}}{2\left(-6\right)}$$

Multiply 24 times 11.

$$t=\frac{-2±\sqrt{4+264}}{2\left(-6\right)}$$

Add 4 to 264.

$$t=\frac{-2±\sqrt{268}}{2\left(-6\right)}$$

Take the square root of 268.

$$t=\frac{-2±2\sqrt{67}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$t=\frac{-2±2\sqrt{67}}{-12}$$

Now solve the equation t=\frac{-2±2\sqrt{67}}{-12} when ± is plus. Add -2 to 2\sqrt{67}.

$$t=\frac{2\sqrt{67}-2}{-12}$$

Divide -2+2\sqrt{67} by -12.

$$t=\frac{1-\sqrt{67}}{6}$$

Now solve the equation t=\frac{-2±2\sqrt{67}}{-12} when ± is minus. Subtract 2\sqrt{67} from -2.

$$t=\frac{-2\sqrt{67}-2}{-12}$$

Divide -2-2\sqrt{67} by -12.

$$t=\frac{\sqrt{67}+1}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1-\sqrt{67}}{6} for x_{1} and \frac{1+\sqrt{67}}{6} for x_{2}.

$$-6t^{2}+2t+11=-6\left(t-\frac{1-\sqrt{67}}{6}\right)\left(t-\frac{\sqrt{67}+1}{6}\right)$$