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13t^{2}-54t-331=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.
t=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 13\left(-331\right)}}{2\times 13}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 13 for a, -54 for b, and -331 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
t=\frac{-\left(-54\right)±\sqrt{2916-4\times 13\left(-331\right)}}{2\times 13}
Square -54.
t=\frac{-\left(-54\right)±\sqrt{2916-52\left(-331\right)}}{2\times 13}
Multiply -4 times 13.
t=\frac{-\left(-54\right)±\sqrt{2916+17212}}{2\times 13}
Multiply -52 times -331.
t=\frac{-\left(-54\right)±\sqrt{20128}}{2\times 13}
Add 2916 to 17212.
t=\frac{-\left(-54\right)±4\sqrt{1258}}{2\times 13}
Take the square root of 20128.
t=\frac{54±4\sqrt{1258}}{2\times 13}
The opposite of -54 is 54.
t=\frac{54±4\sqrt{1258}}{26}
Multiply 2 times 13.
t=\frac{4\sqrt{1258}+54}{26}
Now solve the equation t=\frac{54±4\sqrt{1258}}{26} when ± is plus. Add 54 to 4\sqrt{1258}.
t=\frac{2\sqrt{1258}+27}{13}
Divide 54+4\sqrt{1258} by 26.
t=\frac{54-4\sqrt{1258}}{26}
Now solve the equation t=\frac{54±4\sqrt{1258}}{26} when ± is minus. Subtract 4\sqrt{1258} from 54.
t=\frac{27-2\sqrt{1258}}{13}
Divide 54-4\sqrt{1258} by 26.
t=\frac{2\sqrt{1258}+27}{13} t=\frac{27-2\sqrt{1258}}{13}
The equation is now solved.
13t^{2}-54t-331=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.
13t^{2}-54t-331-\left(-331\right)=-\left(-331\right)
Add 331 to both sides of the equation.
13t^{2}-54t=-\left(-331\right)
Subtracting -331 from itself leaves 0.
13t^{2}-54t=331
Subtract -331 from 0.
\frac{13t^{2}-54t}{13}=\frac{331}{13}
Divide both sides by 13.
t^{2}-\frac{54}{13}t=\frac{331}{13}
Dividing by 13 undoes the multiplication by 13.
t^{2}-\frac{54}{13}t+\left(-\frac{27}{13}\right)^{2}=\frac{331}{13}+\left(-\frac{27}{13}\right)^{2}
Divide -\frac{54}{13}, the coefficient of the x term, by 2 to get -\frac{27}{13}. Then add the square of -\frac{27}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
t^{2}-\frac{54}{13}t+\frac{729}{169}=\frac{331}{13}+\frac{729}{169}
Square -\frac{27}{13} by squaring both the numerator and the denominator of the fraction.
t^{2}-\frac{54}{13}t+\frac{729}{169}=\frac{5032}{169}
Add \frac{331}{13} to \frac{729}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(t-\frac{27}{13}\right)^{2}=\frac{5032}{169}
Factor t^{2}-\frac{54}{13}t+\frac{729}{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(t-\frac{27}{13}\right)^{2}}=\sqrt{\frac{5032}{169}}
Take the square root of both sides of the equation.
t-\frac{27}{13}=\frac{2\sqrt{1258}}{13} t-\frac{27}{13}=-\frac{2\sqrt{1258}}{13}
Simplify.
t=\frac{2\sqrt{1258}+27}{13} t=\frac{27-2\sqrt{1258}}{13}
Add \frac{27}{13} to both sides of the equation.
x ^ 2 -\frac{54}{13}x -\frac{331}{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{54}{13} rs = -\frac{331}{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{27}{13} - u s = \frac{27}{13} + u
Two numbers r and s sum up to \frac{54}{13} exactly when the average of the two numbers is \frac{1}{2}*\frac{54}{13} = \frac{27}{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{27}{13} - u) (\frac{27}{13} + u) = -\frac{331}{13}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{331}{13}
\frac{729}{169} - u^2 = -\frac{331}{13}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{331}{13}-\frac{729}{169} = -\frac{5032}{169}
Simplify the expression by subtracting \frac{729}{169} on both sides
u^2 = \frac{5032}{169} u = \pm\sqrt{\frac{5032}{169}} = \pm \frac{\sqrt{5032}}{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{27}{13} - \frac{\sqrt{5032}}{13} = -3.380 s = \frac{27}{13} + \frac{\sqrt{5032}}{13} = 7.534
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.