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16m^{2}-40m-9=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.
m=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 16\left(-9\right)}}{2\times 16}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -40 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-\left(-40\right)±\sqrt{1600-4\times 16\left(-9\right)}}{2\times 16}
Square -40.
m=\frac{-\left(-40\right)±\sqrt{1600-64\left(-9\right)}}{2\times 16}
Multiply -4 times 16.
m=\frac{-\left(-40\right)±\sqrt{1600+576}}{2\times 16}
Multiply -64 times -9.
m=\frac{-\left(-40\right)±\sqrt{2176}}{2\times 16}
Add 1600 to 576.
m=\frac{-\left(-40\right)±8\sqrt{34}}{2\times 16}
Take the square root of 2176.
m=\frac{40±8\sqrt{34}}{2\times 16}
The opposite of -40 is 40.
m=\frac{40±8\sqrt{34}}{32}
Multiply 2 times 16.
m=\frac{8\sqrt{34}+40}{32}
Now solve the equation m=\frac{40±8\sqrt{34}}{32} when ± is plus. Add 40 to 8\sqrt{34}.
m=\frac{\sqrt{34}+5}{4}
Divide 40+8\sqrt{34} by 32.
m=\frac{40-8\sqrt{34}}{32}
Now solve the equation m=\frac{40±8\sqrt{34}}{32} when ± is minus. Subtract 8\sqrt{34} from 40.
m=\frac{5-\sqrt{34}}{4}
Divide 40-8\sqrt{34} by 32.
m=\frac{\sqrt{34}+5}{4} m=\frac{5-\sqrt{34}}{4}
The equation is now solved.
16m^{2}-40m-9=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.
16m^{2}-40m-9-\left(-9\right)=-\left(-9\right)
Add 9 to both sides of the equation.
16m^{2}-40m=-\left(-9\right)
Subtracting -9 from itself leaves 0.
16m^{2}-40m=9
Subtract -9 from 0.
\frac{16m^{2}-40m}{16}=\frac{9}{16}
Divide both sides by 16.
m^{2}+\left(-\frac{40}{16}\right)m=\frac{9}{16}
Dividing by 16 undoes the multiplication by 16.
m^{2}-\frac{5}{2}m=\frac{9}{16}
Reduce the fraction \frac{-40}{16} to lowest terms by extracting and canceling out 8.
m^{2}-\frac{5}{2}m+\left(-\frac{5}{4}\right)^{2}=\frac{9}{16}+\left(-\frac{5}{4}\right)^{2}
Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
m^{2}-\frac{5}{2}m+\frac{25}{16}=\frac{9+25}{16}
Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.
m^{2}-\frac{5}{2}m+\frac{25}{16}=\frac{17}{8}
Add \frac{9}{16} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(m-\frac{5}{4}\right)^{2}=\frac{17}{8}
Factor m^{2}-\frac{5}{2}m+\frac{25}{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(m-\frac{5}{4}\right)^{2}}=\sqrt{\frac{17}{8}}
Take the square root of both sides of the equation.
m-\frac{5}{4}=\frac{\sqrt{34}}{4} m-\frac{5}{4}=-\frac{\sqrt{34}}{4}
Simplify.
m=\frac{\sqrt{34}+5}{4} m=\frac{5-\sqrt{34}}{4}
Add \frac{5}{4} to both sides of the equation.