Factor
\left(u-2\right)\left(2u+9\right)
Evaluate
\left(u-2\right)\left(2u+9\right)
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a+b=5 ab=2\left(-18\right)=-36
Factor the expression by grouping. First, the expression needs to be rewritten as 2u^{2}+au+bu-18. To find a and b, set up a system to be solved.
-1,36 -2,18 -3,12 -4,9 -6,6
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -36.
-1+36=35 -2+18=16 -3+12=9 -4+9=5 -6+6=0
Calculate the sum for each pair.
a=-4 b=9
The solution is the pair that gives sum 5.
\left(2u^{2}-4u\right)+\left(9u-18\right)
Rewrite 2u^{2}+5u-18 as \left(2u^{2}-4u\right)+\left(9u-18\right).
2u\left(u-2\right)+9\left(u-2\right)
Factor out 2u in the first and 9 in the second group.
\left(u-2\right)\left(2u+9\right)
Factor out common term u-2 by using distributive property.
2u^{2}+5u-18=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{-5±\sqrt{5^{2}-4\times 2\left(-18\right)}}{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.
u=\frac{-5±\sqrt{25-4\times 2\left(-18\right)}}{2\times 2}
Square 5.
u=\frac{-5±\sqrt{25-8\left(-18\right)}}{2\times 2}
Multiply -4 times 2.
u=\frac{-5±\sqrt{25+144}}{2\times 2}
Multiply -8 times -18.
u=\frac{-5±\sqrt{169}}{2\times 2}
Add 25 to 144.
u=\frac{-5±13}{2\times 2}
Take the square root of 169.
u=\frac{-5±13}{4}
Multiply 2 times 2.
u=\frac{8}{4}
Now solve the equation u=\frac{-5±13}{4} when ± is plus. Add -5 to 13.
u=2
Divide 8 by 4.
u=-\frac{18}{4}
Now solve the equation u=\frac{-5±13}{4} when ± is minus. Subtract 13 from -5.
u=-\frac{9}{2}
Reduce the fraction \frac{-18}{4} to lowest terms by extracting and canceling out 2.
2u^{2}+5u-18=2\left(u-2\right)\left(u-\left(-\frac{9}{2}\right)\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{9}{2} for x_{2}.
2u^{2}+5u-18=2\left(u-2\right)\left(u+\frac{9}{2}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
2u^{2}+5u-18=2\left(u-2\right)\times \frac{2u+9}{2}
Add \frac{9}{2} to u by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
2u^{2}+5u-18=\left(u-2\right)\left(2u+9\right)
Cancel out 2, the greatest common factor in 2 and 2.
x ^ 2 +\frac{5}{2}x -9 = 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{5}{2} rs = -9
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{5}{4} - u s = -\frac{5}{4} + u
Two numbers r and s sum up to -\frac{5}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{5}{2} = -\frac{5}{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{5}{4} - u) (-\frac{5}{4} + u) = -9
To solve for unknown quantity u, substitute these in the product equation rs = -9
\frac{25}{16} - u^2 = -9
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -9-\frac{25}{16} = -\frac{169}{16}
Simplify the expression by subtracting \frac{25}{16} on both sides
u^2 = \frac{169}{16} u = \pm\sqrt{\frac{169}{16}} = \pm \frac{13}{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{5}{4} - \frac{13}{4} = -4.500 s = -\frac{5}{4} + \frac{13}{4} = 2
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}