Solve 20-120+9quad16p^2-24p+9)

$$20 - 120 + 9 \quad 16 p ^ { 2 } - 24 p + 9 )$$

Evaluate

$$144p^{2}-24p-91$$

Solution Steps

Subtract 120 from 20 to get -100.

$$-100+9\times 16p^{2}-24p+9$$

Multiply 9 and 16 to get 144.

$$-100+144p^{2}-24p+9$$

Add -100 and 9 to get -91.

$$-91+144p^{2}-24p$$

Factor

$$144\left(p-\left(-\frac{\sqrt{23}}{6}+\frac{1}{12}\right)\right)\left(p-\left(\frac{\sqrt{23}}{6}+\frac{1}{12}\right)\right)$$

Steps Using the Quadratic Formula

Subtract 120 from 20 to get -100.

$$factor(-100+9\times 16p^{2}-24p+9)$$

Multiply 9 and 16 to get 144.

$$factor(-100+144p^{2}-24p+9)$$

Add -100 and 9 to get -91.

$$factor(-91+144p^{2}-24p)$$

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.

$$144p^{2}-24p-91=0$$

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.

$$p=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 144\left(-91\right)}}{2\times 144}$$

Square -24.

$$p=\frac{-\left(-24\right)±\sqrt{576-4\times 144\left(-91\right)}}{2\times 144}$$

Multiply -4 times 144.

$$p=\frac{-\left(-24\right)±\sqrt{576-576\left(-91\right)}}{2\times 144}$$

Multiply -576 times -91.

$$p=\frac{-\left(-24\right)±\sqrt{576+52416}}{2\times 144}$$

Add 576 to 52416.

$$p=\frac{-\left(-24\right)±\sqrt{52992}}{2\times 144}$$

Take the square root of 52992.

$$p=\frac{-\left(-24\right)±48\sqrt{23}}{2\times 144}$$

The opposite of -24 is 24.

$$p=\frac{24±48\sqrt{23}}{2\times 144}$$

Multiply 2 times 144.

$$p=\frac{24±48\sqrt{23}}{288}$$

Now solve the equation p=\frac{24±48\sqrt{23}}{288} when ± is plus. Add 24 to 48\sqrt{23}.

$$p=\frac{48\sqrt{23}+24}{288}$$

Divide 24+48\sqrt{23} by 288.

$$p=\frac{\sqrt{23}}{6}+\frac{1}{12}$$

Now solve the equation p=\frac{24±48\sqrt{23}}{288} when ± is minus. Subtract 48\sqrt{23} from 24.

$$p=\frac{24-48\sqrt{23}}{288}$$

Divide 24-48\sqrt{23} by 288.

$$p=-\frac{\sqrt{23}}{6}+\frac{1}{12}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{12}+\frac{\sqrt{23}}{6} for x_{1} and \frac{1}{12}-\frac{\sqrt{23}}{6} for x_{2}.

$$144p^{2}-24p-91=144\left(p-\left(\frac{\sqrt{23}}{6}+\frac{1}{12}\right)\right)\left(p-\left(-\frac{\sqrt{23}}{6}+\frac{1}{12}\right)\right)$$