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a^{2}-204a-36=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.
a=\frac{-\left(-204\right)±\sqrt{\left(-204\right)^{2}-4\left(-36\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -204 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-\left(-204\right)±\sqrt{41616-4\left(-36\right)}}{2}
Square -204.
a=\frac{-\left(-204\right)±\sqrt{41616+144}}{2}
Multiply -4 times -36.
a=\frac{-\left(-204\right)±\sqrt{41760}}{2}
Add 41616 to 144.
a=\frac{-\left(-204\right)±12\sqrt{290}}{2}
Take the square root of 41760.
a=\frac{204±12\sqrt{290}}{2}
The opposite of -204 is 204.
a=\frac{12\sqrt{290}+204}{2}
Now solve the equation a=\frac{204±12\sqrt{290}}{2} when ± is plus. Add 204 to 12\sqrt{290}.
a=6\sqrt{290}+102
Divide 204+12\sqrt{290} by 2.
a=\frac{204-12\sqrt{290}}{2}
Now solve the equation a=\frac{204±12\sqrt{290}}{2} when ± is minus. Subtract 12\sqrt{290} from 204.
a=102-6\sqrt{290}
Divide 204-12\sqrt{290} by 2.
a=6\sqrt{290}+102 a=102-6\sqrt{290}
The equation is now solved.
a^{2}-204a-36=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.
a^{2}-204a-36-\left(-36\right)=-\left(-36\right)
Add 36 to both sides of the equation.
a^{2}-204a=-\left(-36\right)
Subtracting -36 from itself leaves 0.
a^{2}-204a=36
Subtract -36 from 0.
a^{2}-204a+\left(-102\right)^{2}=36+\left(-102\right)^{2}
Divide -204, the coefficient of the x term, by 2 to get -102. Then add the square of -102 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
a^{2}-204a+10404=36+10404
Square -102.
a^{2}-204a+10404=10440
Add 36 to 10404.
\left(a-102\right)^{2}=10440
Factor a^{2}-204a+10404. 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(a-102\right)^{2}}=\sqrt{10440}
Take the square root of both sides of the equation.
a-102=6\sqrt{290} a-102=-6\sqrt{290}
Simplify.
a=6\sqrt{290}+102 a=102-6\sqrt{290}
Add 102 to both sides of the equation.
x ^ 2 -204x -36 = 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 = 204 rs = -36
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 = 102 - u s = 102 + u
Two numbers r and s sum up to 204 exactly when the average of the two numbers is \frac{1}{2}*204 = 102. 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>
(102 - u) (102 + u) = -36
To solve for unknown quantity u, substitute these in the product equation rs = -36
10404 - u^2 = -36
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -36-10404 = -10440
Simplify the expression by subtracting 10404 on both sides
u^2 = 10440 u = \pm\sqrt{10440} = \pm \sqrt{10440}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =102 - \sqrt{10440} = -0.176 s = 102 + \sqrt{10440} = 204.176
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.