Solve for x
x=20\sqrt{26}-20\approx 81.980390272
x=-20\sqrt{26}-20\approx -121.980390272
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x^{2}+40x-10000=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{-40±\sqrt{40^{2}-4\left(-10000\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 40 for b, and -10000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-40±\sqrt{1600-4\left(-10000\right)}}{2}
Square 40.
x=\frac{-40±\sqrt{1600+40000}}{2}
Multiply -4 times -10000.
x=\frac{-40±\sqrt{41600}}{2}
Add 1600 to 40000.
x=\frac{-40±40\sqrt{26}}{2}
Take the square root of 41600.
x=\frac{40\sqrt{26}-40}{2}
Now solve the equation x=\frac{-40±40\sqrt{26}}{2} when ± is plus. Add -40 to 40\sqrt{26}.
x=20\sqrt{26}-20
Divide -40+40\sqrt{26} by 2.
x=\frac{-40\sqrt{26}-40}{2}
Now solve the equation x=\frac{-40±40\sqrt{26}}{2} when ± is minus. Subtract 40\sqrt{26} from -40.
x=-20\sqrt{26}-20
Divide -40-40\sqrt{26} by 2.
x=20\sqrt{26}-20 x=-20\sqrt{26}-20
The equation is now solved.
x^{2}+40x-10000=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.
x^{2}+40x-10000-\left(-10000\right)=-\left(-10000\right)
Add 10000 to both sides of the equation.
x^{2}+40x=-\left(-10000\right)
Subtracting -10000 from itself leaves 0.
x^{2}+40x=10000
Subtract -10000 from 0.
x^{2}+40x+20^{2}=10000+20^{2}
Divide 40, the coefficient of the x term, by 2 to get 20. Then add the square of 20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+40x+400=10000+400
Square 20.
x^{2}+40x+400=10400
Add 10000 to 400.
\left(x+20\right)^{2}=10400
Factor x^{2}+40x+400. 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+20\right)^{2}}=\sqrt{10400}
Take the square root of both sides of the equation.
x+20=20\sqrt{26} x+20=-20\sqrt{26}
Simplify.
x=20\sqrt{26}-20 x=-20\sqrt{26}-20
Subtract 20 from both sides of the equation.
x ^ 2 +40x -10000 = 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 = -40 rs = -10000
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 = -20 - u s = -20 + u
Two numbers r and s sum up to -40 exactly when the average of the two numbers is \frac{1}{2}*-40 = -20. 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>
(-20 - u) (-20 + u) = -10000
To solve for unknown quantity u, substitute these in the product equation rs = -10000
400 - u^2 = -10000
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -10000-400 = -10400
Simplify the expression by subtracting 400 on both sides
u^2 = 10400 u = \pm\sqrt{10400} = \pm \sqrt{10400}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-20 - \sqrt{10400} = -121.980 s = -20 + \sqrt{10400} = 81.980
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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