Solve -3p^2+4p+8

$$- 3 p ^ { 2 } + 4 p + 8$$

Factor

$$-3\left(p-\frac{2-2\sqrt{7}}{3}\right)\left(p-\frac{2\sqrt{7}+2}{3}\right)$$

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.

$$-3p^{2}+4p+8=0$$

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.

$$p=\frac{-4±\sqrt{4^{2}-4\left(-3\right)\times 8}}{2\left(-3\right)}$$

Square 4.

$$p=\frac{-4±\sqrt{16-4\left(-3\right)\times 8}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$p=\frac{-4±\sqrt{16+12\times 8}}{2\left(-3\right)}$$

Multiply 12 times 8.

$$p=\frac{-4±\sqrt{16+96}}{2\left(-3\right)}$$

Add 16 to 96.

$$p=\frac{-4±\sqrt{112}}{2\left(-3\right)}$$

Take the square root of 112.

$$p=\frac{-4±4\sqrt{7}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$p=\frac{-4±4\sqrt{7}}{-6}$$

Now solve the equation p=\frac{-4±4\sqrt{7}}{-6} when ± is plus. Add -4 to 4\sqrt{7}.

$$p=\frac{4\sqrt{7}-4}{-6}$$

Divide -4+4\sqrt{7} by -6.

$$p=\frac{2-2\sqrt{7}}{3}$$

Now solve the equation p=\frac{-4±4\sqrt{7}}{-6} when ± is minus. Subtract 4\sqrt{7} from -4.

$$p=\frac{-4\sqrt{7}-4}{-6}$$

Divide -4-4\sqrt{7} by -6.

$$p=\frac{2\sqrt{7}+2}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2-2\sqrt{7}}{3} for x_{1} and \frac{2+2\sqrt{7}}{3} for x_{2}.

$$-3p^{2}+4p+8=-3\left(p-\frac{2-2\sqrt{7}}{3}\right)\left(p-\frac{2\sqrt{7}+2}{3}\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -\frac{4}{3}x -\frac{8}{3} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = \frac{4}{3} rs = -\frac{8}{3}$$

Two numbers r and s sum up to \frac{4}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{4}{3} = \frac{2}{3}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{2}{3} - u s = \frac{2}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{8}{3}

$$(\frac{2}{3} - u) (\frac{2}{3} + u) = -\frac{8}{3}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{4}{9} - u^2 = -\frac{8}{3}$$

Simplify the expression by subtracting \frac{4}{9} on both sides

$$-u^2 = -\frac{8}{3}-\frac{4}{9} = -\frac{28}{9}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{28}{9} u = \pm\sqrt{\frac{28}{9}} = \pm \frac{\sqrt{28}}{3}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{2}{3} - \frac{\sqrt{28}}{3} = -1.097 s = \frac{2}{3} + \frac{\sqrt{28}}{3} = 2.431$$

Evaluate

$$8+4p-3p^{2}$$