Solve -4p^2-3p-2+p^2+4

$$- 4 p ^ { 2 } - 3 p - 2 + p ^ { 2 } + 4$$

Evaluate

$$2-3p-3p^{2}$$

Solution Steps

Combine -4p^{2} and p^{2} to get -3p^{2}.

$$-3p^{2}-3p-2+4$$

Add -2 and 4 to get 2.

$$-3p^{2}-3p+2$$

Factor

$$-3\left(p-\left(-\frac{\sqrt{33}}{6}-\frac{1}{2}\right)\right)\left(p-\left(\frac{\sqrt{33}}{6}-\frac{1}{2}\right)\right)$$

Steps Using the Quadratic Formula

Combine -4p^{2} and p^{2} to get -3p^{2}.

$$factor(-3p^{2}-3p-2+4)$$

Add -2 and 4 to get 2.

$$factor(-3p^{2}-3p+2)$$

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.

$$-3p^{2}-3p+2=0$$

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.

$$p=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-3\right)\times 2}}{2\left(-3\right)}$$

Square -3.

$$p=\frac{-\left(-3\right)±\sqrt{9-4\left(-3\right)\times 2}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$p=\frac{-\left(-3\right)±\sqrt{9+12\times 2}}{2\left(-3\right)}$$

Multiply 12 times 2.

$$p=\frac{-\left(-3\right)±\sqrt{9+24}}{2\left(-3\right)}$$

Add 9 to 24.

$$p=\frac{-\left(-3\right)±\sqrt{33}}{2\left(-3\right)}$$

The opposite of -3 is 3.

$$p=\frac{3±\sqrt{33}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$p=\frac{3±\sqrt{33}}{-6}$$

Now solve the equation p=\frac{3±\sqrt{33}}{-6} when ± is plus. Add 3 to \sqrt{33}.

$$p=\frac{\sqrt{33}+3}{-6}$$

Divide 3+\sqrt{33} by -6.

$$p=-\frac{\sqrt{33}}{6}-\frac{1}{2}$$

Now solve the equation p=\frac{3±\sqrt{33}}{-6} when ± is minus. Subtract \sqrt{33} from 3.

$$p=\frac{3-\sqrt{33}}{-6}$$

Divide 3-\sqrt{33} by -6.

$$p=\frac{\sqrt{33}}{6}-\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{2}-\frac{\sqrt{33}}{6} for x_{1} and -\frac{1}{2}+\frac{\sqrt{33}}{6} for x_{2}.

$$-3p^{2}-3p+2=-3\left(p-\left(-\frac{\sqrt{33}}{6}-\frac{1}{2}\right)\right)\left(p-\left(\frac{\sqrt{33}}{6}-\frac{1}{2}\right)\right)$$