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$\exponential{x}{2} - 400 x + 40000 = 0 $
Solve for x
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x^{2}-400x+40000=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.
x=\frac{-\left(-400\right)±\sqrt{\left(-400\right)^{2}-4\times 40000}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -400 for b, and 40000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-400\right)±\sqrt{160000-4\times 40000}}{2}
Square -400.
x=\frac{-\left(-400\right)±\sqrt{160000-160000}}{2}
Multiply -4 times 40000.
x=\frac{-\left(-400\right)±\sqrt{0}}{2}
Add 160000 to -160000.
x=-\frac{-400}{2}
Take the square root of 0.
x=\frac{400}{2}
The opposite of -400 is 400.
x=200
Divide 400 by 2.
x^{2}-400x+40000=0
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.
\left(x-200\right)^{2}=0
Factor x^{2}-400x+40000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-200\right)^{2}}=\sqrt{0}
Take the square root of both sides of the equation.
x-200=0 x-200=0
Simplify.
x=200 x=200
Add 200 to both sides of the equation.
x=200
The equation is now solved. Solutions are the same.
x ^ 2 -400x +40000 = 0
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.
r + s = 400 rs = 40000
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 = 200 - u s = 200 + u
Two numbers r and s sum up to 400 exactly when the average of the two numbers is \frac{1}{2}*400 = 200. 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>
(200 - u) (200 + u) = 40000
To solve for unknown quantity u, substitute these in the product equation rs = 40000
40000 - u^2 = 40000
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 40000-40000 = 0
Simplify the expression by subtracting 40000 on both sides
u^2 = 0 u = 0
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r = s = 200
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.