Solve h^2-12800h+40960000=0

$$h ^ { 2 } - 12800 h + 40960000 = 0$$

Solve for h

$$h=6400$$

Steps Using the Quadratic Formula

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.

$$h^{2}-12800h+40960000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12800 for b, and 40960000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-\left(-12800\right)±\sqrt{\left(-12800\right)^{2}-4\times 40960000}}{2}$$

Square -12800.

$$h=\frac{-\left(-12800\right)±\sqrt{163840000-4\times 40960000}}{2}$$

Multiply -4 times 40960000.

$$h=\frac{-\left(-12800\right)±\sqrt{163840000-163840000}}{2}$$

Add 163840000 to -163840000.

$$h=\frac{-\left(-12800\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$h=-\frac{-12800}{2}$$

The opposite of -12800 is 12800.

$$h=\frac{12800}{2}$$

Divide 12800 by 2.

$$h=6400$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$h^{2}-12800h+40960000=0$$

Factor h^{2}-12800h+40960000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(h-6400\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(h-6400\right)^{2}}=\sqrt{0}$$

Simplify.

$$h-6400=0 h-6400=0$$

Add 6400 to both sides of the equation.

$$h=6400 h=6400$$

The equation is now solved. Solutions are the same.

$$h=6400$$

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.

$$x ^ 2 -12800x [phone] = 0$$

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

$$r + s = 12800 rs = 40960000$$

Two numbers r and s sum up to 12800 exactly when the average of the two numbers is \frac{1}{2}*12800 = 6400. 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>

$$r = 6400 - u s = 6400 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 40960000

$$(6400 - u) (6400 + u) = 40960000$$

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

$$40960000 - u^2 = 40960000$$

Simplify the expression by subtracting 40960000 on both sides

$$-u^2 = 40960000-40960000 = 0$$

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

$$u^2 = 0 u = 0$$

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

$$r = s = 6400$$