Solve for x (complex solution)
x=\frac{5+\sqrt{16999}i}{192}\approx 0.026041667+0.679063611i
x=\frac{-\sqrt{16999}i+5}{192}\approx 0.026041667-0.679063611i
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1152x^{2}-60x+532=0
Swap sides so that all variable terms are on the left hand side.
x=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 1152\times 532}}{2\times 1152}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1152 for a, -60 for b, and 532 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-60\right)±\sqrt{3600-4\times 1152\times 532}}{2\times 1152}
Square -60.
x=\frac{-\left(-60\right)±\sqrt{3600-4608\times 532}}{2\times 1152}
Multiply -4 times 1152.
x=\frac{-\left(-60\right)±\sqrt{3600-2451456}}{2\times 1152}
Multiply -4608 times 532.
x=\frac{-\left(-60\right)±\sqrt{-2447856}}{2\times 1152}
Add 3600 to -2451456.
x=\frac{-\left(-60\right)±12\sqrt{16999}i}{2\times 1152}
Take the square root of -2447856.
x=\frac{60±12\sqrt{16999}i}{2\times 1152}
The opposite of -60 is 60.
x=\frac{60±12\sqrt{16999}i}{2304}
Multiply 2 times 1152.
x=\frac{60+12\sqrt{16999}i}{2304}
Now solve the equation x=\frac{60±12\sqrt{16999}i}{2304} when ± is plus. Add 60 to 12i\sqrt{16999}.
x=\frac{5+\sqrt{16999}i}{192}
Divide 60+12i\sqrt{16999} by 2304.
x=\frac{-12\sqrt{16999}i+60}{2304}
Now solve the equation x=\frac{60±12\sqrt{16999}i}{2304} when ± is minus. Subtract 12i\sqrt{16999} from 60.
x=\frac{-\sqrt{16999}i+5}{192}
Divide 60-12i\sqrt{16999} by 2304.
x=\frac{5+\sqrt{16999}i}{192} x=\frac{-\sqrt{16999}i+5}{192}
The equation is now solved.
1152x^{2}-60x+532=0
Swap sides so that all variable terms are on the left hand side.
1152x^{2}-60x=-532
Subtract 532 from both sides. Anything subtracted from zero gives its negation.
\frac{1152x^{2}-60x}{1152}=-\frac{532}{1152}
Divide both sides by 1152.
x^{2}+\left(-\frac{60}{1152}\right)x=-\frac{532}{1152}
Dividing by 1152 undoes the multiplication by 1152.
x^{2}-\frac{5}{96}x=-\frac{532}{1152}
Reduce the fraction \frac{-60}{1152} to lowest terms by extracting and canceling out 12.
x^{2}-\frac{5}{96}x=-\frac{133}{288}
Reduce the fraction \frac{-532}{1152} to lowest terms by extracting and canceling out 4.
x^{2}-\frac{5}{96}x+\left(-\frac{5}{192}\right)^{2}=-\frac{133}{288}+\left(-\frac{5}{192}\right)^{2}
Divide -\frac{5}{96}, the coefficient of the x term, by 2 to get -\frac{5}{192}. Then add the square of -\frac{5}{192} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{5}{96}x+\frac{25}{36864}=-\frac{133}{288}+\frac{25}{36864}
Square -\frac{5}{192} by squaring both the numerator and the denominator of the fraction.
x^{2}-\frac{5}{96}x+\frac{25}{36864}=-\frac{16999}{36864}
Add -\frac{133}{288} to \frac{25}{36864} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x-\frac{5}{192}\right)^{2}=-\frac{16999}{36864}
Factor x^{2}-\frac{5}{96}x+\frac{25}{36864}. 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(x-\frac{5}{192}\right)^{2}}=\sqrt{-\frac{16999}{36864}}
Take the square root of both sides of the equation.
x-\frac{5}{192}=\frac{\sqrt{16999}i}{192} x-\frac{5}{192}=-\frac{\sqrt{16999}i}{192}
Simplify.
x=\frac{5+\sqrt{16999}i}{192} x=\frac{-\sqrt{16999}i+5}{192}
Add \frac{5}{192} to both sides of the equation.
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