Factor
\left(m-3\right)\left(2m+3\right)
Evaluate
\left(m-3\right)\left(2m+3\right)
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a+b=-3 ab=2\left(-9\right)=-18
Factor the expression by grouping. First, the expression needs to be rewritten as 2m^{2}+am+bm-9. To find a and b, set up a system to be solved.
1,-18 2,-9 3,-6
Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -18.
1-18=-17 2-9=-7 3-6=-3
Calculate the sum for each pair.
a=-6 b=3
The solution is the pair that gives sum -3.
\left(2m^{2}-6m\right)+\left(3m-9\right)
Rewrite 2m^{2}-3m-9 as \left(2m^{2}-6m\right)+\left(3m-9\right).
2m\left(m-3\right)+3\left(m-3\right)
Factor out 2m in the first and 3 in the second group.
\left(m-3\right)\left(2m+3\right)
Factor out common term m-3 by using distributive property.
2m^{2}-3m-9=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
m=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-9\right)}}{2\times 2}
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(-3\right)±\sqrt{9-4\times 2\left(-9\right)}}{2\times 2}
Square -3.
m=\frac{-\left(-3\right)±\sqrt{9-8\left(-9\right)}}{2\times 2}
Multiply -4 times 2.
m=\frac{-\left(-3\right)±\sqrt{9+72}}{2\times 2}
Multiply -8 times -9.
m=\frac{-\left(-3\right)±\sqrt{81}}{2\times 2}
Add 9 to 72.
m=\frac{-\left(-3\right)±9}{2\times 2}
Take the square root of 81.
m=\frac{3±9}{2\times 2}
The opposite of -3 is 3.
m=\frac{3±9}{4}
Multiply 2 times 2.
m=\frac{12}{4}
Now solve the equation m=\frac{3±9}{4} when ± is plus. Add 3 to 9.
m=3
Divide 12 by 4.
m=-\frac{6}{4}
Now solve the equation m=\frac{3±9}{4} when ± is minus. Subtract 9 from 3.
m=-\frac{3}{2}
Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.
2m^{2}-3m-9=2\left(m-3\right)\left(m-\left(-\frac{3}{2}\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and -\frac{3}{2} for x_{2}.
2m^{2}-3m-9=2\left(m-3\right)\left(m+\frac{3}{2}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
2m^{2}-3m-9=2\left(m-3\right)\times \frac{2m+3}{2}
Add \frac{3}{2} to m by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
2m^{2}-3m-9=\left(m-3\right)\left(2m+3\right)
Cancel out 2, the greatest common factor in 2 and 2.
x ^ 2 -\frac{3}{2}x -\frac{9}{2} = 0
Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 2
r + s = \frac{3}{2} rs = -\frac{9}{2}
Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C
r = \frac{3}{4} - u s = \frac{3}{4} + u
Two numbers r and s sum up to \frac{3}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{3}{2} = \frac{3}{4}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>
(\frac{3}{4} - u) (\frac{3}{4} + u) = -\frac{9}{2}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{2}
\frac{9}{16} - u^2 = -\frac{9}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{9}{2}-\frac{9}{16} = -\frac{81}{16}
Simplify the expression by subtracting \frac{9}{16} on both sides
u^2 = \frac{81}{16} u = \pm\sqrt{\frac{81}{16}} = \pm \frac{9}{4}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{3}{4} - \frac{9}{4} = -1.500 s = \frac{3}{4} + \frac{9}{4} = 3
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
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y = 3x + 4
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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 ) }
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Limits
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