Solve 2p^2-2400000p+7000000

$$2 p ^ { 2 } - 2400000 p + 7000000$$

Factor

$$2\left(p-\left(600000-500\sqrt{1439986}\right)\right)\left(p-\left(500\sqrt{1439986}+600000\right)\right)$$

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.

$$2p^{2}-2400000p+7000000=0$$

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.

$$p=\frac{-\left(-2400000\right)±\sqrt{\left(-2400000\right)^{2}-4\times 2\times 7000000}}{2\times 2}$$

Square -2400000.

$$p=\frac{-\left(-2400000\right)±\sqrt{5760000000000-4\times 2\times 7000000}}{2\times 2}$$

Multiply -4 times 2.

$$p=\frac{-\left(-2400000\right)±\sqrt{5760000000000-8\times 7000000}}{2\times 2}$$

Multiply -8 times 7000000.

$$p=\frac{-\left(-2400000\right)±\sqrt{5760000000000-56000000}}{2\times 2}$$

Add 5760000000000 to -56000000.

$$p=\frac{-\left(-2400000\right)±\sqrt{5759944000000}}{2\times 2}$$

Take the square root of 5759944000000.

$$p=\frac{-\left(-2400000\right)±2000\sqrt{1439986}}{2\times 2}$$

The opposite of -2400000 is 2400000.

$$p=\frac{2400000±2000\sqrt{1439986}}{2\times 2}$$

Multiply 2 times 2.

$$p=\frac{2400000±2000\sqrt{1439986}}{4}$$

Now solve the equation p=\frac{2400000±2000\sqrt{1439986}}{4} when ± is plus. Add 2400000 to 2000\sqrt{1439986}.

$$p=\frac{2000\sqrt{1439986}+2400000}{4}$$

Divide 2400000+2000\sqrt{1439986} by 4.

$$p=500\sqrt{1439986}+600000$$

Now solve the equation p=\frac{2400000±2000\sqrt{1439986}}{4} when ± is minus. Subtract 2000\sqrt{1439986} from 2400000.

$$p=\frac{2400000-2000\sqrt{1439986}}{4}$$

Divide 2400000-2000\sqrt{1439986} by 4.

$$p=600000-500\sqrt{1439986}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 600000+500\sqrt{1439986} for x_{1} and 600000-500\sqrt{1439986} for x_{2}.

$$2p^{2}-2400000p+7000000=2\left(p-\left(500\sqrt{1439986}+600000\right)\right)\left(p-\left(600000-500\sqrt{1439986}\right)\right)$$

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 2

$$x ^ 2 -1200000x +3500000 = 0$$

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

$$r + s = 1200000 rs = 3500000$$

Two numbers r and s sum up to 1200000 exactly when the average of the two numbers is \frac{1}{2}*1200000 = 600000. 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>

$$r = 600000 - u s = 600000 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 3500000

$$(600000 - u) (600000 + u) = 3500000$$

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

$$28053504 - u^2 = 3500000$$

Simplify the expression by subtracting 28053504 on both sides

$$-u^2 = 3500000-28053504 = -24553504$$

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

$$u^2 = 24553504 u = \pm\sqrt{24553504} = \pm \sqrt{24553504}$$

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

$$r =600000 - \sqrt{24553504} = 2.917 s = 600000 + \sqrt{24553504} = 1199997.083$$

Evaluate

$$2\left(p^{2}-1200000p+3500000\right)$$