Solve 10p^2-45p-40

$$10 p ^ { 2 } - 45 p - 40$$

Factor

$$10\left(p-\frac{9-\sqrt{145}}{4}\right)\left(p-\frac{\sqrt{145}+9}{4}\right)$$

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.

$$10p^{2}-45p-40=0$$

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.

$$p=\frac{-\left(-45\right)±\sqrt{\left(-45\right)^{2}-4\times 10\left(-40\right)}}{2\times 10}$$

Square -45.

$$p=\frac{-\left(-45\right)±\sqrt{2025-4\times 10\left(-40\right)}}{2\times 10}$$

Multiply -4 times 10.

$$p=\frac{-\left(-45\right)±\sqrt{2025-40\left(-40\right)}}{2\times 10}$$

Multiply -40 times -40.

$$p=\frac{-\left(-45\right)±\sqrt{2025+1600}}{2\times 10}$$

Add 2025 to 1600.

$$p=\frac{-\left(-45\right)±\sqrt{3625}}{2\times 10}$$

Take the square root of 3625.

$$p=\frac{-\left(-45\right)±5\sqrt{145}}{2\times 10}$$

The opposite of -45 is 45.

$$p=\frac{45±5\sqrt{145}}{2\times 10}$$

Multiply 2 times 10.

$$p=\frac{45±5\sqrt{145}}{20}$$

Now solve the equation p=\frac{45±5\sqrt{145}}{20} when ± is plus. Add 45 to 5\sqrt{145}.

$$p=\frac{5\sqrt{145}+45}{20}$$

Divide 45+5\sqrt{145} by 20.

$$p=\frac{\sqrt{145}+9}{4}$$

Now solve the equation p=\frac{45±5\sqrt{145}}{20} when ± is minus. Subtract 5\sqrt{145} from 45.

$$p=\frac{45-5\sqrt{145}}{20}$$

Divide 45-5\sqrt{145} by 20.

$$p=\frac{9-\sqrt{145}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9+\sqrt{145}}{4} for x_{1} and \frac{9-\sqrt{145}}{4} for x_{2}.

$$10p^{2}-45p-40=10\left(p-\frac{\sqrt{145}+9}{4}\right)\left(p-\frac{9-\sqrt{145}}{4}\right)$$

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.This is achieved by dividing both sides of the equation by 10

$$x ^ 2 -\frac{9}{2}x -4 = 0$$

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

$$r + s = \frac{9}{2} rs = -4$$

Two numbers r and s sum up to \frac{9}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{9}{2} = \frac{9}{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>

$$r = \frac{9}{4} - u s = \frac{9}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -4

$$(\frac{9}{4} - u) (\frac{9}{4} + u) = -4$$

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

$$\frac{81}{16} - u^2 = -4$$

Simplify the expression by subtracting \frac{81}{16} on both sides

$$-u^2 = -4-\frac{81}{16} = -\frac{145}{16}$$

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

$$u^2 = \frac{145}{16} u = \pm\sqrt{\frac{145}{16}} = \pm \frac{\sqrt{145}}{4}$$

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

$$r =\frac{9}{4} - \frac{\sqrt{145}}{4} = -0.760 s = \frac{9}{4} + \frac{\sqrt{145}}{4} = 5.260$$

Evaluate

$$10p^{2}-45p-40$$