Solve 15p^2+24p+8

$$15 p ^ { 2 } + 24 p + 8$$

Factor

$$15\left(p-\left(-\frac{2\sqrt{6}}{15}-\frac{4}{5}\right)\right)\left(p-\left(\frac{2\sqrt{6}}{15}-\frac{4}{5}\right)\right)$$

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.

$$15p^{2}+24p+8=0$$

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.

$$p=\frac{-24±\sqrt{24^{2}-4\times 15\times 8}}{2\times 15}$$

Square 24.

$$p=\frac{-24±\sqrt{576-4\times 15\times 8}}{2\times 15}$$

Multiply -4 times 15.

$$p=\frac{-24±\sqrt{576-60\times 8}}{2\times 15}$$

Multiply -60 times 8.

$$p=\frac{-24±\sqrt{576-480}}{2\times 15}$$

Add 576 to -480.

$$p=\frac{-24±\sqrt{96}}{2\times 15}$$

Take the square root of 96.

$$p=\frac{-24±4\sqrt{6}}{2\times 15}$$

Multiply 2 times 15.

$$p=\frac{-24±4\sqrt{6}}{30}$$

Now solve the equation p=\frac{-24±4\sqrt{6}}{30} when ± is plus. Add -24 to 4\sqrt{6}.

$$p=\frac{4\sqrt{6}-24}{30}$$

Divide -24+4\sqrt{6} by 30.

$$p=\frac{2\sqrt{6}}{15}-\frac{4}{5}$$

Now solve the equation p=\frac{-24±4\sqrt{6}}{30} when ± is minus. Subtract 4\sqrt{6} from -24.

$$p=\frac{-4\sqrt{6}-24}{30}$$

Divide -24-4\sqrt{6} by 30.

$$p=-\frac{2\sqrt{6}}{15}-\frac{4}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{4}{5}+\frac{2\sqrt{6}}{15} for x_{1} and -\frac{4}{5}-\frac{2\sqrt{6}}{15} for x_{2}.

$$15p^{2}+24p+8=15\left(p-\left(\frac{2\sqrt{6}}{15}-\frac{4}{5}\right)\right)\left(p-\left(-\frac{2\sqrt{6}}{15}-\frac{4}{5}\right)\right)$$

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.This is achieved by dividing both sides of the equation by 15

$$x ^ 2 +\frac{8}{5}x +\frac{8}{15} = 0$$

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

$$r + s = -\frac{8}{5} rs = \frac{8}{15}$$

Two numbers r and s sum up to -\frac{8}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{5} = -\frac{4}{5}. 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>

$$r = -\frac{4}{5} - u s = -\frac{4}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{15}

$$(-\frac{4}{5} - u) (-\frac{4}{5} + u) = \frac{8}{15}$$

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

$$\frac{16}{25} - u^2 = \frac{8}{15}$$

Simplify the expression by subtracting \frac{16}{25} on both sides

$$-u^2 = \frac{8}{15}-\frac{16}{25} = -\frac{8}{75}$$

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

$$u^2 = \frac{8}{75} u = \pm\sqrt{\frac{8}{75}} = \pm \frac{\sqrt{8}}{\sqrt{75}}$$

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

$$r =-\frac{4}{5} - \frac{\sqrt{8}}{\sqrt{75}} = -1.127 s = -\frac{4}{5} + \frac{\sqrt{8}}{\sqrt{75}} = -0.473$$

Evaluate

$$15p^{2}+24p+8$$