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a+b=-4 ab=5\left(-9\right)=-45
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5m^{2}+am+bm-9. To find a and b, set up a system to be solved.
1,-45 3,-15 5,-9
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 -45.
1-45=-44 3-15=-12 5-9=-4
Calculate the sum for each pair.
a=-9 b=5
The solution is the pair that gives sum -4.
\left(5m^{2}-9m\right)+\left(5m-9\right)
Rewrite 5m^{2}-4m-9 as \left(5m^{2}-9m\right)+\left(5m-9\right).
m\left(5m-9\right)+5m-9
Factor out m in 5m^{2}-9m.
\left(5m-9\right)\left(m+1\right)
Factor out common term 5m-9 by using distributive property.
m=\frac{9}{5} m=-1
To find equation solutions, solve 5m-9=0 and m+1=0.
5m^{2}-4m-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(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 5\left(-9\right)}}{2\times 5}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -4 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-\left(-4\right)±\sqrt{16-4\times 5\left(-9\right)}}{2\times 5}
Square -4.
m=\frac{-\left(-4\right)±\sqrt{16-20\left(-9\right)}}{2\times 5}
Multiply -4 times 5.
m=\frac{-\left(-4\right)±\sqrt{16+180}}{2\times 5}
Multiply -20 times -9.
m=\frac{-\left(-4\right)±\sqrt{196}}{2\times 5}
Add 16 to 180.
m=\frac{-\left(-4\right)±14}{2\times 5}
Take the square root of 196.
m=\frac{4±14}{2\times 5}
The opposite of -4 is 4.
m=\frac{4±14}{10}
Multiply 2 times 5.
m=\frac{18}{10}
Now solve the equation m=\frac{4±14}{10} when ± is plus. Add 4 to 14.
m=\frac{9}{5}
Reduce the fraction \frac{18}{10} to lowest terms by extracting and canceling out 2.
m=-\frac{10}{10}
Now solve the equation m=\frac{4±14}{10} when ± is minus. Subtract 14 from 4.
m=-1
Divide -10 by 10.
m=\frac{9}{5} m=-1
The equation is now solved.
5m^{2}-4m-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.
5m^{2}-4m-9-\left(-9\right)=-\left(-9\right)
Add 9 to both sides of the equation.
5m^{2}-4m=-\left(-9\right)
Subtracting -9 from itself leaves 0.
5m^{2}-4m=9
Subtract -9 from 0.
\frac{5m^{2}-4m}{5}=\frac{9}{5}
Divide both sides by 5.
m^{2}-\frac{4}{5}m=\frac{9}{5}
Dividing by 5 undoes the multiplication by 5.
m^{2}-\frac{4}{5}m+\left(-\frac{2}{5}\right)^{2}=\frac{9}{5}+\left(-\frac{2}{5}\right)^{2}
Divide -\frac{4}{5}, the coefficient of the x term, by 2 to get -\frac{2}{5}. Then add the square of -\frac{2}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
m^{2}-\frac{4}{5}m+\frac{4}{25}=\frac{9}{5}+\frac{4}{25}
Square -\frac{2}{5} by squaring both the numerator and the denominator of the fraction.
m^{2}-\frac{4}{5}m+\frac{4}{25}=\frac{49}{25}
Add \frac{9}{5} to \frac{4}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(m-\frac{2}{5}\right)^{2}=\frac{49}{25}
Factor m^{2}-\frac{4}{5}m+\frac{4}{25}. 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{2}{5}\right)^{2}}=\sqrt{\frac{49}{25}}
Take the square root of both sides of the equation.
m-\frac{2}{5}=\frac{7}{5} m-\frac{2}{5}=-\frac{7}{5}
Simplify.
m=\frac{9}{5} m=-1
Add \frac{2}{5} to both sides of the equation.
x ^ 2 -\frac{4}{5}x -\frac{9}{5} = 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 5
r + s = \frac{4}{5} rs = -\frac{9}{5}
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{2}{5} - u s = \frac{2}{5} + u
Two numbers r and s sum up to \frac{4}{5} exactly when the average of the two numbers is \frac{1}{2}*\frac{4}{5} = \frac{2}{5}. 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{2}{5} - u) (\frac{2}{5} + u) = -\frac{9}{5}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{5}
\frac{4}{25} - u^2 = -\frac{9}{5}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{9}{5}-\frac{4}{25} = -\frac{49}{25}
Simplify the expression by subtracting \frac{4}{25} on both sides
u^2 = \frac{49}{25} u = \pm\sqrt{\frac{49}{25}} = \pm \frac{7}{5}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{2}{5} - \frac{7}{5} = -1.000 s = \frac{2}{5} + \frac{7}{5} = 1.800
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.