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a+b=-8 ab=-5\times 4=-20
Factor the expression by grouping. First, the expression needs to be rewritten as -5u^{2}+au+bu+4. To find a and b, set up a system to be solved.
1,-20 2,-10 4,-5
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 -20.
1-20=-19 2-10=-8 4-5=-1
Calculate the sum for each pair.
a=2 b=-10
The solution is the pair that gives sum -8.
\left(-5u^{2}+2u\right)+\left(-10u+4\right)
Rewrite -5u^{2}-8u+4 as \left(-5u^{2}+2u\right)+\left(-10u+4\right).
-u\left(5u-2\right)-2\left(5u-2\right)
Factor out -u in the first and -2 in the second group.
\left(5u-2\right)\left(-u-2\right)
Factor out common term 5u-2 by using distributive property.
-5u^{2}-8u+4=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.
u=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-5\right)\times 4}}{2\left(-5\right)}
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.
u=\frac{-\left(-8\right)±\sqrt{64-4\left(-5\right)\times 4}}{2\left(-5\right)}
Square -8.
u=\frac{-\left(-8\right)±\sqrt{64+20\times 4}}{2\left(-5\right)}
Multiply -4 times -5.
u=\frac{-\left(-8\right)±\sqrt{64+80}}{2\left(-5\right)}
Multiply 20 times 4.
u=\frac{-\left(-8\right)±\sqrt{144}}{2\left(-5\right)}
Add 64 to 80.
u=\frac{-\left(-8\right)±12}{2\left(-5\right)}
Take the square root of 144.
u=\frac{8±12}{2\left(-5\right)}
The opposite of -8 is 8.
u=\frac{8±12}{-10}
Multiply 2 times -5.
u=\frac{20}{-10}
Now solve the equation u=\frac{8±12}{-10} when ± is plus. Add 8 to 12.
u=-2
Divide 20 by -10.
u=-\frac{4}{-10}
Now solve the equation u=\frac{8±12}{-10} when ± is minus. Subtract 12 from 8.
u=\frac{2}{5}
Reduce the fraction \frac{-4}{-10} to lowest terms by extracting and canceling out 2.
-5u^{2}-8u+4=-5\left(u-\left(-2\right)\right)\left(u-\frac{2}{5}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and \frac{2}{5} for x_{2}.
-5u^{2}-8u+4=-5\left(u+2\right)\left(u-\frac{2}{5}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
-5u^{2}-8u+4=-5\left(u+2\right)\times \frac{-5u+2}{-5}
Subtract \frac{2}{5} from u by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
-5u^{2}-8u+4=\left(u+2\right)\left(-5u+2\right)
Cancel out 5, the greatest common factor in -5 and 5.
x ^ 2 +\frac{8}{5}x -\frac{4}{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.
r + s = -\frac{8}{5} rs = -\frac{4}{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{4}{5} - u s = -\frac{4}{5} + u
Two numbers r and s sum up to -\frac{8}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{8}{5} = -\frac{4}{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{4}{5} - u) (-\frac{4}{5} + u) = -\frac{4}{5}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{4}{5}
\frac{16}{25} - u^2 = -\frac{4}{5}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{4}{5}-\frac{16}{25} = -\frac{36}{25}
Simplify the expression by subtracting \frac{16}{25} on both sides
u^2 = \frac{36}{25} u = \pm\sqrt{\frac{36}{25}} = \pm \frac{6}{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{4}{5} - \frac{6}{5} = -2 s = -\frac{4}{5} + \frac{6}{5} = 0.400
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.