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-5n^{2}+251n-3020=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.
n=\frac{-251±\sqrt{251^{2}-4\left(-5\right)\left(-3020\right)}}{2\left(-5\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 251 for b, and -3020 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-251±\sqrt{63001-4\left(-5\right)\left(-3020\right)}}{2\left(-5\right)}
Square 251.
n=\frac{-251±\sqrt{63001+20\left(-3020\right)}}{2\left(-5\right)}
Multiply -4 times -5.
n=\frac{-251±\sqrt{63001-60400}}{2\left(-5\right)}
Multiply 20 times -3020.
n=\frac{-251±\sqrt{2601}}{2\left(-5\right)}
Add 63001 to -60400.
n=\frac{-251±51}{2\left(-5\right)}
Take the square root of 2601.
n=\frac{-251±51}{-10}
Multiply 2 times -5.
n=-\frac{200}{-10}
Now solve the equation n=\frac{-251±51}{-10} when ± is plus. Add -251 to 51.
n=20
Divide -200 by -10.
n=-\frac{302}{-10}
Now solve the equation n=\frac{-251±51}{-10} when ± is minus. Subtract 51 from -251.
n=\frac{151}{5}
Reduce the fraction \frac{-302}{-10} to lowest terms by extracting and canceling out 2.
n=20 n=\frac{151}{5}
The equation is now solved.
-5n^{2}+251n-3020=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.
-5n^{2}+251n-3020-\left(-3020\right)=-\left(-3020\right)
Add 3020 to both sides of the equation.
-5n^{2}+251n=-\left(-3020\right)
Subtracting -3020 from itself leaves 0.
-5n^{2}+251n=3020
Subtract -3020 from 0.
\frac{-5n^{2}+251n}{-5}=\frac{3020}{-5}
Divide both sides by -5.
n^{2}+\frac{251}{-5}n=\frac{3020}{-5}
Dividing by -5 undoes the multiplication by -5.
n^{2}-\frac{251}{5}n=\frac{3020}{-5}
Divide 251 by -5.
n^{2}-\frac{251}{5}n=-604
Divide 3020 by -5.
n^{2}-\frac{251}{5}n+\left(-\frac{251}{10}\right)^{2}=-604+\left(-\frac{251}{10}\right)^{2}
Divide -\frac{251}{5}, the coefficient of the x term, by 2 to get -\frac{251}{10}. Then add the square of -\frac{251}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
n^{2}-\frac{251}{5}n+\frac{63001}{100}=-604+\frac{63001}{100}
Square -\frac{251}{10} by squaring both the numerator and the denominator of the fraction.
n^{2}-\frac{251}{5}n+\frac{63001}{100}=\frac{2601}{100}
Add -604 to \frac{63001}{100}.
\left(n-\frac{251}{10}\right)^{2}=\frac{2601}{100}
Factor n^{2}-\frac{251}{5}n+\frac{63001}{100}. 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(n-\frac{251}{10}\right)^{2}}=\sqrt{\frac{2601}{100}}
Take the square root of both sides of the equation.
n-\frac{251}{10}=\frac{51}{10} n-\frac{251}{10}=-\frac{51}{10}
Simplify.
n=\frac{151}{5} n=20
Add \frac{251}{10} to both sides of the equation.
x ^ 2 -\frac{251}{5}x +604 = 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 = \frac{251}{5} rs = 604
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{251}{10} - u s = \frac{251}{10} + u
Two numbers r and s sum up to \frac{251}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{251}{5} = \frac{251}{10}. 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{251}{10} - u) (\frac{251}{10} + u) = 604
To solve for unknown quantity u, substitute these in the product equation rs = 604
\frac{63001}{100} - u^2 = 604
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 604-\frac{63001}{100} = -\frac{2601}{100}
Simplify the expression by subtracting \frac{63001}{100} on both sides
u^2 = \frac{2601}{100} u = \pm\sqrt{\frac{2601}{100}} = \pm \frac{51}{10}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{251}{10} - \frac{51}{10} = 20.000 s = \frac{251}{10} + \frac{51}{10} = 30.200
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.