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4\left(-2x^{2}-3x+5\right)
Factor out 4.
a+b=-3 ab=-2\times 5=-10
Consider -2x^{2}-3x+5. Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx+5. To find a and b, set up a system to be solved.
1,-10 2,-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 -10.
1-10=-9 2-5=-3
Calculate the sum for each pair.
a=2 b=-5
The solution is the pair that gives sum -3.
\left(-2x^{2}+2x\right)+\left(-5x+5\right)
Rewrite -2x^{2}-3x+5 as \left(-2x^{2}+2x\right)+\left(-5x+5\right).
2x\left(-x+1\right)+5\left(-x+1\right)
Factor out 2x in the first and 5 in the second group.
\left(-x+1\right)\left(2x+5\right)
Factor out common term -x+1 by using distributive property.
4\left(-x+1\right)\left(2x+5\right)
Rewrite the complete factored expression.
-8x^{2}-12x+20=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.
x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-8\right)\times 20}}{2\left(-8\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.
x=\frac{-\left(-12\right)±\sqrt{144-4\left(-8\right)\times 20}}{2\left(-8\right)}
Square -12.
x=\frac{-\left(-12\right)±\sqrt{144+32\times 20}}{2\left(-8\right)}
Multiply -4 times -8.
x=\frac{-\left(-12\right)±\sqrt{144+640}}{2\left(-8\right)}
Multiply 32 times 20.
x=\frac{-\left(-12\right)±\sqrt{784}}{2\left(-8\right)}
Add 144 to 640.
x=\frac{-\left(-12\right)±28}{2\left(-8\right)}
Take the square root of 784.
x=\frac{12±28}{2\left(-8\right)}
The opposite of -12 is 12.
x=\frac{12±28}{-16}
Multiply 2 times -8.
x=\frac{40}{-16}
Now solve the equation x=\frac{12±28}{-16} when ± is plus. Add 12 to 28.
x=-\frac{5}{2}
Reduce the fraction \frac{40}{-16} to lowest terms by extracting and canceling out 8.
x=-\frac{16}{-16}
Now solve the equation x=\frac{12±28}{-16} when ± is minus. Subtract 28 from 12.
x=1
Divide -16 by -16.
-8x^{2}-12x+20=-8\left(x-\left(-\frac{5}{2}\right)\right)\left(x-1\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{5}{2} for x_{1} and 1 for x_{2}.
-8x^{2}-12x+20=-8\left(x+\frac{5}{2}\right)\left(x-1\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
-8x^{2}-12x+20=-8\times \frac{-2x-5}{-2}\left(x-1\right)
Add \frac{5}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
-8x^{2}-12x+20=4\left(-2x-5\right)\left(x-1\right)
Cancel out 2, the greatest common factor in -8 and 2.
x ^ 2 +\frac{3}{2}x -\frac{5}{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.
r + s = -\frac{3}{2} rs = -\frac{5}{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{5}{2}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{2}
\frac{9}{16} - u^2 = -\frac{5}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{5}{2}-\frac{9}{16} = -\frac{49}{16}
Simplify the expression by subtracting \frac{9}{16} on both sides
u^2 = \frac{49}{16} u = \pm\sqrt{\frac{49}{16}} = \pm \frac{7}{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{7}{4} = -2.500 s = -\frac{3}{4} + \frac{7}{4} = 1
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.