Solve for w
w=-\frac{21}{47}\approx -0.446808511
w=0
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w\left(94w+42\right)=0
Factor out w.
w=0 w=-\frac{21}{47}
To find equation solutions, solve w=0 and 94w+42=0.
94w^{2}+42w=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{-42±\sqrt{42^{2}}}{2\times 94}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 94 for a, 42 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
w=\frac{-42±42}{2\times 94}
Take the square root of 42^{2}.
w=\frac{-42±42}{188}
Multiply 2 times 94.
w=\frac{0}{188}
Now solve the equation w=\frac{-42±42}{188} when ± is plus. Add -42 to 42.
w=0
Divide 0 by 188.
w=-\frac{84}{188}
Now solve the equation w=\frac{-42±42}{188} when ± is minus. Subtract 42 from -42.
w=-\frac{21}{47}
Reduce the fraction \frac{-84}{188} to lowest terms by extracting and canceling out 4.
w=0 w=-\frac{21}{47}
The equation is now solved.
94w^{2}+42w=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{94w^{2}+42w}{94}=\frac{0}{94}
Divide both sides by 94.
w^{2}+\frac{42}{94}w=\frac{0}{94}
Dividing by 94 undoes the multiplication by 94.
w^{2}+\frac{21}{47}w=\frac{0}{94}
Reduce the fraction \frac{42}{94} to lowest terms by extracting and canceling out 2.
w^{2}+\frac{21}{47}w=0
Divide 0 by 94.
w^{2}+\frac{21}{47}w+\left(\frac{21}{94}\right)^{2}=\left(\frac{21}{94}\right)^{2}
Divide \frac{21}{47}, the coefficient of the x term, by 2 to get \frac{21}{94}. Then add the square of \frac{21}{94} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
w^{2}+\frac{21}{47}w+\frac{441}{8836}=\frac{441}{8836}
Square \frac{21}{94} by squaring both the numerator and the denominator of the fraction.
\left(w+\frac{21}{94}\right)^{2}=\frac{441}{8836}
Factor w^{2}+\frac{21}{47}w+\frac{441}{8836}. 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{21}{94}\right)^{2}}=\sqrt{\frac{441}{8836}}
Take the square root of both sides of the equation.
w+\frac{21}{94}=\frac{21}{94} w+\frac{21}{94}=-\frac{21}{94}
Simplify.
w=0 w=-\frac{21}{47}
Subtract \frac{21}{94} from both sides of the equation.
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