Solve D(p)=-2p^2-3p+900

$$D ( p ) = - 2 p ^ { 2 } - 3 p + 900$$

Factor

$$-2\left(p-\frac{-9\sqrt{89}-3}{4}\right)\left(p-\frac{9\sqrt{89}-3}{4}\right)$$

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.

$$-2p^{2}-3p+900=0$$

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.

$$p=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-2\right)\times 900}}{2\left(-2\right)}$$

Square -3.

$$p=\frac{-\left(-3\right)±\sqrt{9-4\left(-2\right)\times 900}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$p=\frac{-\left(-3\right)±\sqrt{9+8\times 900}}{2\left(-2\right)}$$

Multiply 8 times 900.

$$p=\frac{-\left(-3\right)±\sqrt{9+7200}}{2\left(-2\right)}$$

Add 9 to 7200.

$$p=\frac{-\left(-3\right)±\sqrt{7209}}{2\left(-2\right)}$$

Take the square root of 7209.

$$p=\frac{-\left(-3\right)±9\sqrt{89}}{2\left(-2\right)}$$

The opposite of -3 is 3.

$$p=\frac{3±9\sqrt{89}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$p=\frac{3±9\sqrt{89}}{-4}$$

Now solve the equation p=\frac{3±9\sqrt{89}}{-4} when ± is plus. Add 3 to 9\sqrt{89}.

$$p=\frac{9\sqrt{89}+3}{-4}$$

Divide 3+9\sqrt{89} by -4.

$$p=\frac{-9\sqrt{89}-3}{4}$$

Now solve the equation p=\frac{3±9\sqrt{89}}{-4} when ± is minus. Subtract 9\sqrt{89} from 3.

$$p=\frac{3-9\sqrt{89}}{-4}$$

Divide 3-9\sqrt{89} by -4.

$$p=\frac{9\sqrt{89}-3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3-9\sqrt{89}}{4} for x_{1} and \frac{-3+9\sqrt{89}}{4} for x_{2}.

$$-2p^{2}-3p+900=-2\left(p-\frac{-9\sqrt{89}-3}{4}\right)\left(p-\frac{9\sqrt{89}-3}{4}\right)$$

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.

$$x ^ 2 +\frac{3}{2}x -450 = 0$$

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

$$r + s = -\frac{3}{2} rs = -450$$

Two numbers r and s sum up to -\frac{3}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{3}{2} = -\frac{3}{4}. 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>

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

To solve for unknown quantity u, substitute these in the product equation rs = -450

$$(-\frac{3}{4} - u) (-\frac{3}{4} + u) = -450$$

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

$$\frac{9}{16} - u^2 = -450$$

Simplify the expression by subtracting \frac{9}{16} on both sides

$$-u^2 = -450-\frac{9}{16} = -\frac{7209}{16}$$

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

$$u^2 = \frac{7209}{16} u = \pm\sqrt{\frac{7209}{16}} = \pm \frac{\sqrt{7209}}{4}$$

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

$$r =-\frac{3}{4} - \frac{\sqrt{7209}}{4} = -21.976 s = -\frac{3}{4} + \frac{\sqrt{7209}}{4} = 20.476$$

Evaluate

$$900-3p-2p^{2}$$