Solve 8p-13+11p^2

$$8 p - 13 + 11 p ^ { 2 }$$

Factor

$$11\left(p-\frac{-\sqrt{159}-4}{11}\right)\left(p-\frac{\sqrt{159}-4}{11}\right)$$

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.

$$11p^{2}+8p-13=0$$

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.

$$p=\frac{-8±\sqrt{8^{2}-4\times 11\left(-13\right)}}{2\times 11}$$

Square 8.

$$p=\frac{-8±\sqrt{64-4\times 11\left(-13\right)}}{2\times 11}$$

Multiply -4 times 11.

$$p=\frac{-8±\sqrt{64-44\left(-13\right)}}{2\times 11}$$

Multiply -44 times -13.

$$p=\frac{-8±\sqrt{64+572}}{2\times 11}$$

Add 64 to 572.

$$p=\frac{-8±\sqrt{636}}{2\times 11}$$

Take the square root of 636.

$$p=\frac{-8±2\sqrt{159}}{2\times 11}$$

Multiply 2 times 11.

$$p=\frac{-8±2\sqrt{159}}{22}$$

Now solve the equation p=\frac{-8±2\sqrt{159}}{22} when ± is plus. Add -8 to 2\sqrt{159}.

$$p=\frac{2\sqrt{159}-8}{22}$$

Divide -8+2\sqrt{159} by 22.

$$p=\frac{\sqrt{159}-4}{11}$$

Now solve the equation p=\frac{-8±2\sqrt{159}}{22} when ± is minus. Subtract 2\sqrt{159} from -8.

$$p=\frac{-2\sqrt{159}-8}{22}$$

Divide -8-2\sqrt{159} by 22.

$$p=\frac{-\sqrt{159}-4}{11}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-4+\sqrt{159}}{11} for x_{1} and \frac{-4-\sqrt{159}}{11} for x_{2}.

$$11p^{2}+8p-13=11\left(p-\frac{\sqrt{159}-4}{11}\right)\left(p-\frac{-\sqrt{159}-4}{11}\right)$$

Evaluate

$$11p^{2}+8p-13$$