Solve for x (complex solution)
x=70+10\sqrt{7}i\approx 70+26.457513111i
x=-10\sqrt{7}i+70\approx 70-26.457513111i
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x^{2}-140x+5600=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(-140\right)±\sqrt{\left(-140\right)^{2}-4\times 5600}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -140 for b, and 5600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-140\right)±\sqrt{19600-4\times 5600}}{2}
Square -140.
x=\frac{-\left(-140\right)±\sqrt{19600-22400}}{2}
Multiply -4 times 5600.
x=\frac{-\left(-140\right)±\sqrt{-2800}}{2}
Add 19600 to -22400.
x=\frac{-\left(-140\right)±20\sqrt{7}i}{2}
Take the square root of -2800.
x=\frac{140±20\sqrt{7}i}{2}
The opposite of -140 is 140.
x=\frac{140+20\sqrt{7}i}{2}
Now solve the equation x=\frac{140±20\sqrt{7}i}{2} when ± is plus. Add 140 to 20i\sqrt{7}.
x=70+10\sqrt{7}i
Divide 140+20i\sqrt{7} by 2.
x=\frac{-20\sqrt{7}i+140}{2}
Now solve the equation x=\frac{140±20\sqrt{7}i}{2} when ± is minus. Subtract 20i\sqrt{7} from 140.
x=-10\sqrt{7}i+70
Divide 140-20i\sqrt{7} by 2.
x=70+10\sqrt{7}i x=-10\sqrt{7}i+70
The equation is now solved.
x^{2}-140x+5600=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}-140x+5600-5600=-5600
Subtract 5600 from both sides of the equation.
x^{2}-140x=-5600
Subtracting 5600 from itself leaves 0.
x^{2}-140x+\left(-70\right)^{2}=-5600+\left(-70\right)^{2}
Divide -140, the coefficient of the x term, by 2 to get -70. Then add the square of -70 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-140x+4900=-5600+4900
Square -70.
x^{2}-140x+4900=-700
Add -5600 to 4900.
\left(x-70\right)^{2}=-700
Factor x^{2}-140x+4900. 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-70\right)^{2}}=\sqrt{-700}
Take the square root of both sides of the equation.
x-70=10\sqrt{7}i x-70=-10\sqrt{7}i
Simplify.
x=70+10\sqrt{7}i x=-10\sqrt{7}i+70
Add 70 to both sides of the equation.
x ^ 2 -140x +5600 = 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 = 140 rs = 5600
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 = 70 - u s = 70 + u
Two numbers r and s sum up to 140 exactly when the average of the two numbers is \frac{1}{2}*140 = 70. 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>
(70 - u) (70 + u) = 5600
To solve for unknown quantity u, substitute these in the product equation rs = 5600
4900 - u^2 = 5600
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 5600-4900 = 700
Simplify the expression by subtracting 4900 on both sides
u^2 = -700 u = \pm\sqrt{-700} = \pm \sqrt{700}i
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =70 - \sqrt{700}i s = 70 + \sqrt{700}i
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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