Solve for w
w=\frac{1}{4}=0.25
w=0
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w\left(-8w+2\right)=0
Factor out w.
w=0 w=\frac{1}{4}
To find equation solutions, solve w=0 and -8w+2=0.
-8w^{2}+2w=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.
w=\frac{-2±\sqrt{2^{2}}}{2\left(-8\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
w=\frac{-2±2}{2\left(-8\right)}
Take the square root of 2^{2}.
w=\frac{-2±2}{-16}
Multiply 2 times -8.
w=\frac{0}{-16}
Now solve the equation w=\frac{-2±2}{-16} when ± is plus. Add -2 to 2.
w=0
Divide 0 by -16.
w=-\frac{4}{-16}
Now solve the equation w=\frac{-2±2}{-16} when ± is minus. Subtract 2 from -2.
w=\frac{1}{4}
Reduce the fraction \frac{-4}{-16} to lowest terms by extracting and canceling out 4.
w=0 w=\frac{1}{4}
The equation is now solved.
-8w^{2}+2w=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{-8w^{2}+2w}{-8}=\frac{0}{-8}
Divide both sides by -8.
w^{2}+\frac{2}{-8}w=\frac{0}{-8}
Dividing by -8 undoes the multiplication by -8.
w^{2}-\frac{1}{4}w=\frac{0}{-8}
Reduce the fraction \frac{2}{-8} to lowest terms by extracting and canceling out 2.
w^{2}-\frac{1}{4}w=0
Divide 0 by -8.
w^{2}-\frac{1}{4}w+\left(-\frac{1}{8}\right)^{2}=\left(-\frac{1}{8}\right)^{2}
Divide -\frac{1}{4}, the coefficient of the x term, by 2 to get -\frac{1}{8}. Then add the square of -\frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
w^{2}-\frac{1}{4}w+\frac{1}{64}=\frac{1}{64}
Square -\frac{1}{8} by squaring both the numerator and the denominator of the fraction.
\left(w-\frac{1}{8}\right)^{2}=\frac{1}{64}
Factor w^{2}-\frac{1}{4}w+\frac{1}{64}. 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(w-\frac{1}{8}\right)^{2}}=\sqrt{\frac{1}{64}}
Take the square root of both sides of the equation.
w-\frac{1}{8}=\frac{1}{8} w-\frac{1}{8}=-\frac{1}{8}
Simplify.
w=\frac{1}{4} w=0
Add \frac{1}{8} to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
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Matrix
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Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}