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n^{2}+n-270=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{-1±\sqrt{1^{2}-4\left(-270\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -270 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-1±\sqrt{1-4\left(-270\right)}}{2}
Square 1.
n=\frac{-1±\sqrt{1+1080}}{2}
Multiply -4 times -270.
n=\frac{-1±\sqrt{1081}}{2}
Add 1 to 1080.
n=\frac{\sqrt{1081}-1}{2}
Now solve the equation n=\frac{-1±\sqrt{1081}}{2} when ± is plus. Add -1 to \sqrt{1081}.
n=\frac{-\sqrt{1081}-1}{2}
Now solve the equation n=\frac{-1±\sqrt{1081}}{2} when ± is minus. Subtract \sqrt{1081} from -1.
n=\frac{\sqrt{1081}-1}{2} n=\frac{-\sqrt{1081}-1}{2}
The equation is now solved.
n^{2}+n-270=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.
n^{2}+n-270-\left(-270\right)=-\left(-270\right)
Add 270 to both sides of the equation.
n^{2}+n=-\left(-270\right)
Subtracting -270 from itself leaves 0.
n^{2}+n=270
Subtract -270 from 0.
n^{2}+n+\left(\frac{1}{2}\right)^{2}=270+\left(\frac{1}{2}\right)^{2}
Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
n^{2}+n+\frac{1}{4}=270+\frac{1}{4}
Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.
n^{2}+n+\frac{1}{4}=\frac{1081}{4}
Add 270 to \frac{1}{4}.
\left(n+\frac{1}{2}\right)^{2}=\frac{1081}{4}
Factor n^{2}+n+\frac{1}{4}. 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{1}{2}\right)^{2}}=\sqrt{\frac{1081}{4}}
Take the square root of both sides of the equation.
n+\frac{1}{2}=\frac{\sqrt{1081}}{2} n+\frac{1}{2}=-\frac{\sqrt{1081}}{2}
Simplify.
n=\frac{\sqrt{1081}-1}{2} n=\frac{-\sqrt{1081}-1}{2}
Subtract \frac{1}{2} from both sides of the equation.
x ^ 2 +1x -270 = 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 = -1 rs = -270
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{1}{2} - u s = -\frac{1}{2} + u
Two numbers r and s sum up to -1 exactly when the average of the two numbers is \frac{1}{2}*-1 = -\frac{1}{2}. 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{1}{2} - u) (-\frac{1}{2} + u) = -270
To solve for unknown quantity u, substitute these in the product equation rs = -270
\frac{1}{4} - u^2 = -270
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -270-\frac{1}{4} = -\frac{1081}{4}
Simplify the expression by subtracting \frac{1}{4} on both sides
u^2 = \frac{1081}{4} u = \pm\sqrt{\frac{1081}{4}} = \pm \frac{\sqrt{1081}}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{1}{2} - \frac{\sqrt{1081}}{2} = -16.939 s = -\frac{1}{2} + \frac{\sqrt{1081}}{2} = 15.939
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.