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13x^{2}-66x+36=0
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.
x=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\times 13\times 36}}{2\times 13}
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(-66\right)±\sqrt{4356-4\times 13\times 36}}{2\times 13}
Square -66.
x=\frac{-\left(-66\right)±\sqrt{4356-52\times 36}}{2\times 13}
Multiply -4 times 13.
x=\frac{-\left(-66\right)±\sqrt{4356-1872}}{2\times 13}
Multiply -52 times 36.
x=\frac{-\left(-66\right)±\sqrt{2484}}{2\times 13}
Add 4356 to -1872.
x=\frac{-\left(-66\right)±6\sqrt{69}}{2\times 13}
Take the square root of 2484.
x=\frac{66±6\sqrt{69}}{2\times 13}
The opposite of -66 is 66.
x=\frac{66±6\sqrt{69}}{26}
Multiply 2 times 13.
x=\frac{6\sqrt{69}+66}{26}
Now solve the equation x=\frac{66±6\sqrt{69}}{26} when ± is plus. Add 66 to 6\sqrt{69}.
x=\frac{3\sqrt{69}+33}{13}
Divide 66+6\sqrt{69} by 26.
x=\frac{66-6\sqrt{69}}{26}
Now solve the equation x=\frac{66±6\sqrt{69}}{26} when ± is minus. Subtract 6\sqrt{69} from 66.
x=\frac{33-3\sqrt{69}}{13}
Divide 66-6\sqrt{69} by 26.
13x^{2}-66x+36=13\left(x-\frac{3\sqrt{69}+33}{13}\right)\left(x-\frac{33-3\sqrt{69}}{13}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{33+3\sqrt{69}}{13} for x_{1} and \frac{33-3\sqrt{69}}{13} for x_{2}.
x ^ 2 -\frac{66}{13}x +\frac{36}{13} = 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.This is achieved by dividing both sides of the equation by 13
r + s = \frac{66}{13} rs = \frac{36}{13}
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 = \frac{33}{13} - u s = \frac{33}{13} + u
Two numbers r and s sum up to \frac{66}{13} exactly when the average of the two numbers is \frac{1}{2}*\frac{66}{13} = \frac{33}{13}. 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>
(\frac{33}{13} - u) (\frac{33}{13} + u) = \frac{36}{13}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{36}{13}
\frac{1089}{169} - u^2 = \frac{36}{13}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{36}{13}-\frac{1089}{169} = -\frac{621}{169}
Simplify the expression by subtracting \frac{1089}{169} on both sides
u^2 = \frac{621}{169} u = \pm\sqrt{\frac{621}{169}} = \pm \frac{\sqrt{621}}{13}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{33}{13} - \frac{\sqrt{621}}{13} = 0.622 s = \frac{33}{13} + \frac{\sqrt{621}}{13} = 4.455
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.