Factor
\left(x-5\right)\left(2x-9\right)
Evaluate
\left(x-5\right)\left(2x-9\right)
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a+b=-19 ab=2\times 45=90
Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+45. To find a and b, set up a system to be solved.
-1,-90 -2,-45 -3,-30 -5,-18 -6,-15 -9,-10
Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 90.
-1-90=-91 -2-45=-47 -3-30=-33 -5-18=-23 -6-15=-21 -9-10=-19
Calculate the sum for each pair.
a=-10 b=-9
The solution is the pair that gives sum -19.
\left(2x^{2}-10x\right)+\left(-9x+45\right)
Rewrite 2x^{2}-19x+45 as \left(2x^{2}-10x\right)+\left(-9x+45\right).
2x\left(x-5\right)-9\left(x-5\right)
Factor out 2x in the first and -9 in the second group.
\left(x-5\right)\left(2x-9\right)
Factor out common term x-5 by using distributive property.
2x^{2}-19x+45=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(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 2\times 45}}{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.
x=\frac{-\left(-19\right)±\sqrt{361-4\times 2\times 45}}{2\times 2}
Square -19.
x=\frac{-\left(-19\right)±\sqrt{361-8\times 45}}{2\times 2}
Multiply -4 times 2.
x=\frac{-\left(-19\right)±\sqrt{361-360}}{2\times 2}
Multiply -8 times 45.
x=\frac{-\left(-19\right)±\sqrt{1}}{2\times 2}
Add 361 to -360.
x=\frac{-\left(-19\right)±1}{2\times 2}
Take the square root of 1.
x=\frac{19±1}{2\times 2}
The opposite of -19 is 19.
x=\frac{19±1}{4}
Multiply 2 times 2.
x=\frac{20}{4}
Now solve the equation x=\frac{19±1}{4} when ± is plus. Add 19 to 1.
x=5
Divide 20 by 4.
x=\frac{18}{4}
Now solve the equation x=\frac{19±1}{4} when ± is minus. Subtract 1 from 19.
x=\frac{9}{2}
Reduce the fraction \frac{18}{4} to lowest terms by extracting and canceling out 2.
2x^{2}-19x+45=2\left(x-5\right)\left(x-\frac{9}{2}\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and \frac{9}{2} for x_{2}.
2x^{2}-19x+45=2\left(x-5\right)\times \frac{2x-9}{2}
Subtract \frac{9}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
2x^{2}-19x+45=\left(x-5\right)\left(2x-9\right)
Cancel out 2, the greatest common factor in 2 and 2.
x ^ 2 -\frac{19}{2}x +\frac{45}{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{19}{2} rs = \frac{45}{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{19}{4} - u s = \frac{19}{4} + u
Two numbers r and s sum up to \frac{19}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{19}{2} = \frac{19}{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{19}{4} - u) (\frac{19}{4} + u) = \frac{45}{2}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{45}{2}
\frac{361}{16} - u^2 = \frac{45}{2}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{45}{2}-\frac{361}{16} = -\frac{1}{16}
Simplify the expression by subtracting \frac{361}{16} on both sides
u^2 = \frac{1}{16} u = \pm\sqrt{\frac{1}{16}} = \pm \frac{1}{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{19}{4} - \frac{1}{4} = 4.500 s = \frac{19}{4} + \frac{1}{4} = 5
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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