Solve for x
x=\frac{9}{64}=0.140625
x=0
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x\left(36-256x\right)=0
Factor out x.
x=0 x=\frac{9}{64}
To find equation solutions, solve x=0 and 36-256x=0.
-256x^{2}+36x=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.
x=\frac{-36±\sqrt{36^{2}}}{2\left(-256\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -256 for a, 36 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-36±36}{2\left(-256\right)}
Take the square root of 36^{2}.
x=\frac{-36±36}{-512}
Multiply 2 times -256.
x=\frac{0}{-512}
Now solve the equation x=\frac{-36±36}{-512} when ± is plus. Add -36 to 36.
x=0
Divide 0 by -512.
x=-\frac{72}{-512}
Now solve the equation x=\frac{-36±36}{-512} when ± is minus. Subtract 36 from -36.
x=\frac{9}{64}
Reduce the fraction \frac{-72}{-512} to lowest terms by extracting and canceling out 8.
x=0 x=\frac{9}{64}
The equation is now solved.
-256x^{2}+36x=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.
\frac{-256x^{2}+36x}{-256}=\frac{0}{-256}
Divide both sides by -256.
x^{2}+\frac{36}{-256}x=\frac{0}{-256}
Dividing by -256 undoes the multiplication by -256.
x^{2}-\frac{9}{64}x=\frac{0}{-256}
Reduce the fraction \frac{36}{-256} to lowest terms by extracting and canceling out 4.
x^{2}-\frac{9}{64}x=0
Divide 0 by -256.
x^{2}-\frac{9}{64}x+\left(-\frac{9}{128}\right)^{2}=\left(-\frac{9}{128}\right)^{2}
Divide -\frac{9}{64}, the coefficient of the x term, by 2 to get -\frac{9}{128}. Then add the square of -\frac{9}{128} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-\frac{9}{64}x+\frac{81}{16384}=\frac{81}{16384}
Square -\frac{9}{128} by squaring both the numerator and the denominator of the fraction.
\left(x-\frac{9}{128}\right)^{2}=\frac{81}{16384}
Factor x^{2}-\frac{9}{64}x+\frac{81}{16384}. 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{9}{128}\right)^{2}}=\sqrt{\frac{81}{16384}}
Take the square root of both sides of the equation.
x-\frac{9}{128}=\frac{9}{128} x-\frac{9}{128}=-\frac{9}{128}
Simplify.
x=\frac{9}{64} x=0
Add \frac{9}{128} to both sides of the equation.
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Simultaneous equation
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Integration
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Limits
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