Solve for a
a = \frac{2 \sqrt{10} + 14}{13} \approx 1.563427332
a=\frac{14-2\sqrt{10}}{13}\approx 0.590418822
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13a^{2}-28a+12=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(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 13\times 12}}{2\times 13}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 13 for a, -28 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-\left(-28\right)±\sqrt{784-4\times 13\times 12}}{2\times 13}
Square -28.
a=\frac{-\left(-28\right)±\sqrt{784-52\times 12}}{2\times 13}
Multiply -4 times 13.
a=\frac{-\left(-28\right)±\sqrt{784-624}}{2\times 13}
Multiply -52 times 12.
a=\frac{-\left(-28\right)±\sqrt{160}}{2\times 13}
Add 784 to -624.
a=\frac{-\left(-28\right)±4\sqrt{10}}{2\times 13}
Take the square root of 160.
a=\frac{28±4\sqrt{10}}{2\times 13}
The opposite of -28 is 28.
a=\frac{28±4\sqrt{10}}{26}
Multiply 2 times 13.
a=\frac{4\sqrt{10}+28}{26}
Now solve the equation a=\frac{28±4\sqrt{10}}{26} when ± is plus. Add 28 to 4\sqrt{10}.
a=\frac{2\sqrt{10}+14}{13}
Divide 28+4\sqrt{10} by 26.
a=\frac{28-4\sqrt{10}}{26}
Now solve the equation a=\frac{28±4\sqrt{10}}{26} when ± is minus. Subtract 4\sqrt{10} from 28.
a=\frac{14-2\sqrt{10}}{13}
Divide 28-4\sqrt{10} by 26.
a=\frac{2\sqrt{10}+14}{13} a=\frac{14-2\sqrt{10}}{13}
The equation is now solved.
13a^{2}-28a+12=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.
13a^{2}-28a+12-12=-12
Subtract 12 from both sides of the equation.
13a^{2}-28a=-12
Subtracting 12 from itself leaves 0.
\frac{13a^{2}-28a}{13}=-\frac{12}{13}
Divide both sides by 13.
a^{2}-\frac{28}{13}a=-\frac{12}{13}
Dividing by 13 undoes the multiplication by 13.
a^{2}-\frac{28}{13}a+\left(-\frac{14}{13}\right)^{2}=-\frac{12}{13}+\left(-\frac{14}{13}\right)^{2}
Divide -\frac{28}{13}, the coefficient of the x term, by 2 to get -\frac{14}{13}. Then add the square of -\frac{14}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
a^{2}-\frac{28}{13}a+\frac{196}{169}=-\frac{12}{13}+\frac{196}{169}
Square -\frac{14}{13} by squaring both the numerator and the denominator of the fraction.
a^{2}-\frac{28}{13}a+\frac{196}{169}=\frac{40}{169}
Add -\frac{12}{13} to \frac{196}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(a-\frac{14}{13}\right)^{2}=\frac{40}{169}
Factor a^{2}-\frac{28}{13}a+\frac{196}{169}. 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-\frac{14}{13}\right)^{2}}=\sqrt{\frac{40}{169}}
Take the square root of both sides of the equation.
a-\frac{14}{13}=\frac{2\sqrt{10}}{13} a-\frac{14}{13}=-\frac{2\sqrt{10}}{13}
Simplify.
a=\frac{2\sqrt{10}+14}{13} a=\frac{14-2\sqrt{10}}{13}
Add \frac{14}{13} to both sides of the equation.
x ^ 2 -\frac{28}{13}x +\frac{12}{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{28}{13} rs = \frac{12}{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{14}{13} - u s = \frac{14}{13} + u
Two numbers r and s sum up to \frac{28}{13} exactly when the average of the two numbers is \frac{1}{2}*\frac{28}{13} = \frac{14}{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{14}{13} - u) (\frac{14}{13} + u) = \frac{12}{13}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{12}{13}
\frac{196}{169} - u^2 = \frac{12}{13}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{12}{13}-\frac{196}{169} = -\frac{40}{169}
Simplify the expression by subtracting \frac{196}{169} on both sides
u^2 = \frac{40}{169} u = \pm\sqrt{\frac{40}{169}} = \pm \frac{\sqrt{40}}{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{14}{13} - \frac{\sqrt{40}}{13} = 0.590 s = \frac{14}{13} + \frac{\sqrt{40}}{13} = 1.563
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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