Solve -t^2+6t+9-18+11

$$- t ^ { 2 } + 6 t + 9 - 18 + 11$$

Evaluate

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

Solution Steps

Subtract 18 from 9 to get -9.

$$-t^{2}+6t-9+11$$

Add -9 and 11 to get 2.

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

Factor

$$-\left(t-\left(3-\sqrt{11}\right)\right)\left(t-\left(\sqrt{11}+3\right)\right)$$

Steps Using the Quadratic Formula

Subtract 18 from 9 to get -9.

$$factor(-t^{2}+6t-9+11)$$

Add -9 and 11 to get 2.

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

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.

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

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.

$$t=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 2}}{2\left(-1\right)}$$

Square 6.

$$t=\frac{-6±\sqrt{36-4\left(-1\right)\times 2}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$t=\frac{-6±\sqrt{36+4\times 2}}{2\left(-1\right)}$$

Multiply 4 times 2.

$$t=\frac{-6±\sqrt{36+8}}{2\left(-1\right)}$$

Add 36 to 8.

$$t=\frac{-6±\sqrt{44}}{2\left(-1\right)}$$

Take the square root of 44.

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

Multiply 2 times -1.

$$t=\frac{-6±2\sqrt{11}}{-2}$$

Now solve the equation t=\frac{-6±2\sqrt{11}}{-2} when ± is plus. Add -6 to 2\sqrt{11}.

$$t=\frac{2\sqrt{11}-6}{-2}$$

Divide -6+2\sqrt{11} by -2.

$$t=3-\sqrt{11}$$

Now solve the equation t=\frac{-6±2\sqrt{11}}{-2} when ± is minus. Subtract 2\sqrt{11} from -6.

$$t=\frac{-2\sqrt{11}-6}{-2}$$

Divide -6-2\sqrt{11} by -2.

$$t=\sqrt{11}+3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3-\sqrt{11} for x_{1} and 3+\sqrt{11} for x_{2}.

$$-t^{2}+6t+2=-\left(t-\left(3-\sqrt{11}\right)\right)\left(t-\left(\sqrt{11}+3\right)\right)$$