Solve for m
m=4
m=5
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2\left(4-m\right)=-\frac{1}{2}m^{2}+\frac{5}{2}m-2
Multiply both sides of the equation by 4, the least common multiple of 2,4.
8-2m=-\frac{1}{2}m^{2}+\frac{5}{2}m-2
Use the distributive property to multiply 2 by 4-m.
8-2m+\frac{1}{2}m^{2}=\frac{5}{2}m-2
Add \frac{1}{2}m^{2} to both sides.
8-2m+\frac{1}{2}m^{2}-\frac{5}{2}m=-2
Subtract \frac{5}{2}m from both sides.
8-\frac{9}{2}m+\frac{1}{2}m^{2}=-2
Combine -2m and -\frac{5}{2}m to get -\frac{9}{2}m.
8-\frac{9}{2}m+\frac{1}{2}m^{2}+2=0
Add 2 to both sides.
10-\frac{9}{2}m+\frac{1}{2}m^{2}=0
Add 8 and 2 to get 10.
\frac{1}{2}m^{2}-\frac{9}{2}m+10=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(-\frac{9}{2}\right)±\sqrt{\left(-\frac{9}{2}\right)^{2}-4\times \frac{1}{2}\times 10}}{2\times \frac{1}{2}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -\frac{9}{2} for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-4\times \frac{1}{2}\times 10}}{2\times \frac{1}{2}}
Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.
m=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-2\times 10}}{2\times \frac{1}{2}}
Multiply -4 times \frac{1}{2}.
m=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{81}{4}-20}}{2\times \frac{1}{2}}
Multiply -2 times 10.
m=\frac{-\left(-\frac{9}{2}\right)±\sqrt{\frac{1}{4}}}{2\times \frac{1}{2}}
Add \frac{81}{4} to -20.
m=\frac{-\left(-\frac{9}{2}\right)±\frac{1}{2}}{2\times \frac{1}{2}}
Take the square root of \frac{1}{4}.
m=\frac{\frac{9}{2}±\frac{1}{2}}{2\times \frac{1}{2}}
The opposite of -\frac{9}{2} is \frac{9}{2}.
m=\frac{\frac{9}{2}±\frac{1}{2}}{1}
Multiply 2 times \frac{1}{2}.
m=\frac{5}{1}
Now solve the equation m=\frac{\frac{9}{2}±\frac{1}{2}}{1} when ± is plus. Add \frac{9}{2} to \frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
m=5
Divide 5 by 1.
m=\frac{4}{1}
Now solve the equation m=\frac{\frac{9}{2}±\frac{1}{2}}{1} when ± is minus. Subtract \frac{1}{2} from \frac{9}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
m=4
Divide 4 by 1.
m=5 m=4
The equation is now solved.
2\left(4-m\right)=-\frac{1}{2}m^{2}+\frac{5}{2}m-2
Multiply both sides of the equation by 4, the least common multiple of 2,4.
8-2m=-\frac{1}{2}m^{2}+\frac{5}{2}m-2
Use the distributive property to multiply 2 by 4-m.
8-2m+\frac{1}{2}m^{2}=\frac{5}{2}m-2
Add \frac{1}{2}m^{2} to both sides.
8-2m+\frac{1}{2}m^{2}-\frac{5}{2}m=-2
Subtract \frac{5}{2}m from both sides.
8-\frac{9}{2}m+\frac{1}{2}m^{2}=-2
Combine -2m and -\frac{5}{2}m to get -\frac{9}{2}m.
-\frac{9}{2}m+\frac{1}{2}m^{2}=-2-8
Subtract 8 from both sides.
-\frac{9}{2}m+\frac{1}{2}m^{2}=-10
Subtract 8 from -2 to get -10.
\frac{1}{2}m^{2}-\frac{9}{2}m=-10
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{\frac{1}{2}m^{2}-\frac{9}{2}m}{\frac{1}{2}}=-\frac{10}{\frac{1}{2}}
Multiply both sides by 2.
m^{2}+\left(-\frac{\frac{9}{2}}{\frac{1}{2}}\right)m=-\frac{10}{\frac{1}{2}}
Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.
m^{2}-9m=-\frac{10}{\frac{1}{2}}
Divide -\frac{9}{2} by \frac{1}{2} by multiplying -\frac{9}{2} by the reciprocal of \frac{1}{2}.
m^{2}-9m=-20
Divide -10 by \frac{1}{2} by multiplying -10 by the reciprocal of \frac{1}{2}.
m^{2}-9m+\left(-\frac{9}{2}\right)^{2}=-20+\left(-\frac{9}{2}\right)^{2}
Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
m^{2}-9m+\frac{81}{4}=-20+\frac{81}{4}
Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.
m^{2}-9m+\frac{81}{4}=\frac{1}{4}
Add -20 to \frac{81}{4}.
\left(m-\frac{9}{2}\right)^{2}=\frac{1}{4}
Factor m^{2}-9m+\frac{81}{4}. 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{9}{2}\right)^{2}}=\sqrt{\frac{1}{4}}
Take the square root of both sides of the equation.
m-\frac{9}{2}=\frac{1}{2} m-\frac{9}{2}=-\frac{1}{2}
Simplify.
m=5 m=4
Add \frac{9}{2} to both sides of the equation.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
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Linear equation
y = 3x + 4
Arithmetic
699 * 533
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
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