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a+b=16 ab=-15\left(-1\right)=15
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -15m^{2}+am+bm-1. To find a and b, set up a system to be solved.
1,15 3,5
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 15.
1+15=16 3+5=8
Calculate the sum for each pair.
a=15 b=1
The solution is the pair that gives sum 16.
\left(-15m^{2}+15m\right)+\left(m-1\right)
Rewrite -15m^{2}+16m-1 as \left(-15m^{2}+15m\right)+\left(m-1\right).
15m\left(-m+1\right)-\left(-m+1\right)
Factor out 15m in the first and -1 in the second group.
\left(-m+1\right)\left(15m-1\right)
Factor out common term -m+1 by using distributive property.
m=1 m=\frac{1}{15}
To find equation solutions, solve -m+1=0 and 15m-1=0.
-15m^{2}+16m-1=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{-16±\sqrt{16^{2}-4\left(-15\right)\left(-1\right)}}{2\left(-15\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 16 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-16±\sqrt{256-4\left(-15\right)\left(-1\right)}}{2\left(-15\right)}
Square 16.
m=\frac{-16±\sqrt{256+60\left(-1\right)}}{2\left(-15\right)}
Multiply -4 times -15.
m=\frac{-16±\sqrt{256-60}}{2\left(-15\right)}
Multiply 60 times -1.
m=\frac{-16±\sqrt{196}}{2\left(-15\right)}
Add 256 to -60.
m=\frac{-16±14}{2\left(-15\right)}
Take the square root of 196.
m=\frac{-16±14}{-30}
Multiply 2 times -15.
m=-\frac{2}{-30}
Now solve the equation m=\frac{-16±14}{-30} when ± is plus. Add -16 to 14.
m=\frac{1}{15}
Reduce the fraction \frac{-2}{-30} to lowest terms by extracting and canceling out 2.
m=-\frac{30}{-30}
Now solve the equation m=\frac{-16±14}{-30} when ± is minus. Subtract 14 from -16.
m=1
Divide -30 by -30.
m=\frac{1}{15} m=1
The equation is now solved.
-15m^{2}+16m-1=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.
-15m^{2}+16m-1-\left(-1\right)=-\left(-1\right)
Add 1 to both sides of the equation.
-15m^{2}+16m=-\left(-1\right)
Subtracting -1 from itself leaves 0.
-15m^{2}+16m=1
Subtract -1 from 0.
\frac{-15m^{2}+16m}{-15}=\frac{1}{-15}
Divide both sides by -15.
m^{2}+\frac{16}{-15}m=\frac{1}{-15}
Dividing by -15 undoes the multiplication by -15.
m^{2}-\frac{16}{15}m=\frac{1}{-15}
Divide 16 by -15.
m^{2}-\frac{16}{15}m=-\frac{1}{15}
Divide 1 by -15.
m^{2}-\frac{16}{15}m+\left(-\frac{8}{15}\right)^{2}=-\frac{1}{15}+\left(-\frac{8}{15}\right)^{2}
Divide -\frac{16}{15}, the coefficient of the x term, by 2 to get -\frac{8}{15}. Then add the square of -\frac{8}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
m^{2}-\frac{16}{15}m+\frac{64}{225}=-\frac{1}{15}+\frac{64}{225}
Square -\frac{8}{15} by squaring both the numerator and the denominator of the fraction.
m^{2}-\frac{16}{15}m+\frac{64}{225}=\frac{49}{225}
Add -\frac{1}{15} to \frac{64}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(m-\frac{8}{15}\right)^{2}=\frac{49}{225}
Factor m^{2}-\frac{16}{15}m+\frac{64}{225}. 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{8}{15}\right)^{2}}=\sqrt{\frac{49}{225}}
Take the square root of both sides of the equation.
m-\frac{8}{15}=\frac{7}{15} m-\frac{8}{15}=-\frac{7}{15}
Simplify.
m=1 m=\frac{1}{15}
Add \frac{8}{15} to both sides of the equation.
x ^ 2 -\frac{16}{15}x +\frac{1}{15} = 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.
r + s = \frac{16}{15} rs = \frac{1}{15}
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{8}{15} - u s = \frac{8}{15} + u
Two numbers r and s sum up to \frac{16}{15} exactly when the average of the two numbers is \frac{1}{2}*\frac{16}{15} = \frac{8}{15}. 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{8}{15} - u) (\frac{8}{15} + u) = \frac{1}{15}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{1}{15}
\frac{64}{225} - u^2 = \frac{1}{15}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{1}{15}-\frac{64}{225} = -\frac{49}{225}
Simplify the expression by subtracting \frac{64}{225} on both sides
u^2 = \frac{49}{225} u = \pm\sqrt{\frac{49}{225}} = \pm \frac{7}{15}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{8}{15} - \frac{7}{15} = 0.067 s = \frac{8}{15} + \frac{7}{15} = 1
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.