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-4n^{2}+2n-3=1
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.
-4n^{2}+2n-3-1=1-1
Subtract 1 from both sides of the equation.
-4n^{2}+2n-3-1=0
Subtracting 1 from itself leaves 0.
-4n^{2}+2n-4=0
Subtract 1 from -3.
n=\frac{-2±\sqrt{2^{2}-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 2 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
n=\frac{-2±\sqrt{4-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}
Square 2.
n=\frac{-2±\sqrt{4+16\left(-4\right)}}{2\left(-4\right)}
Multiply -4 times -4.
n=\frac{-2±\sqrt{4-64}}{2\left(-4\right)}
Multiply 16 times -4.
n=\frac{-2±\sqrt{-60}}{2\left(-4\right)}
Add 4 to -64.
n=\frac{-2±2\sqrt{15}i}{2\left(-4\right)}
Take the square root of -60.
n=\frac{-2±2\sqrt{15}i}{-8}
Multiply 2 times -4.
n=\frac{-2+2\sqrt{15}i}{-8}
Now solve the equation n=\frac{-2±2\sqrt{15}i}{-8} when ± is plus. Add -2 to 2i\sqrt{15}.
n=\frac{-\sqrt{15}i+1}{4}
Divide -2+2i\sqrt{15} by -8.
n=\frac{-2\sqrt{15}i-2}{-8}
Now solve the equation n=\frac{-2±2\sqrt{15}i}{-8} when ± is minus. Subtract 2i\sqrt{15} from -2.
n=\frac{1+\sqrt{15}i}{4}
Divide -2-2i\sqrt{15} by -8.
n=\frac{-\sqrt{15}i+1}{4} n=\frac{1+\sqrt{15}i}{4}
The equation is now solved.
-4n^{2}+2n-3=1
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.
-4n^{2}+2n-3-\left(-3\right)=1-\left(-3\right)
Add 3 to both sides of the equation.
-4n^{2}+2n=1-\left(-3\right)
Subtracting -3 from itself leaves 0.
-4n^{2}+2n=4
Subtract -3 from 1.
\frac{-4n^{2}+2n}{-4}=\frac{4}{-4}
Divide both sides by -4.
n^{2}+\frac{2}{-4}n=\frac{4}{-4}
Dividing by -4 undoes the multiplication by -4.
n^{2}-\frac{1}{2}n=\frac{4}{-4}
Reduce the fraction \frac{2}{-4} to lowest terms by extracting and canceling out 2.
n^{2}-\frac{1}{2}n=-1
Divide 4 by -4.
n^{2}-\frac{1}{2}n+\left(-\frac{1}{4}\right)^{2}=-1+\left(-\frac{1}{4}\right)^{2}
Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
n^{2}-\frac{1}{2}n+\frac{1}{16}=-1+\frac{1}{16}
Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.
n^{2}-\frac{1}{2}n+\frac{1}{16}=-\frac{15}{16}
Add -1 to \frac{1}{16}.
\left(n-\frac{1}{4}\right)^{2}=-\frac{15}{16}
Factor n^{2}-\frac{1}{2}n+\frac{1}{16}. 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}{4}\right)^{2}}=\sqrt{-\frac{15}{16}}
Take the square root of both sides of the equation.
n-\frac{1}{4}=\frac{\sqrt{15}i}{4} n-\frac{1}{4}=-\frac{\sqrt{15}i}{4}
Simplify.
n=\frac{1+\sqrt{15}i}{4} n=\frac{-\sqrt{15}i+1}{4}
Add \frac{1}{4} to both sides of the equation.