Factor
\left(2v-1\right)\left(4v+11\right)
Evaluate
\left(2v-1\right)\left(4v+11\right)
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a+b=18 ab=8\left(-11\right)=-88
Factor the expression by grouping. First, the expression needs to be rewritten as 8v^{2}+av+bv-11. To find a and b, set up a system to be solved.
-1,88 -2,44 -4,22 -8,11
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 -88.
-1+88=87 -2+44=42 -4+22=18 -8+11=3
Calculate the sum for each pair.
a=-4 b=22
The solution is the pair that gives sum 18.
\left(8v^{2}-4v\right)+\left(22v-11\right)
Rewrite 8v^{2}+18v-11 as \left(8v^{2}-4v\right)+\left(22v-11\right).
4v\left(2v-1\right)+11\left(2v-1\right)
Factor out 4v in the first and 11 in the second group.
\left(2v-1\right)\left(4v+11\right)
Factor out common term 2v-1 by using distributive property.
8v^{2}+18v-11=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.
v=\frac{-18±\sqrt{18^{2}-4\times 8\left(-11\right)}}{2\times 8}
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.
v=\frac{-18±\sqrt{324-4\times 8\left(-11\right)}}{2\times 8}
Square 18.
v=\frac{-18±\sqrt{324-32\left(-11\right)}}{2\times 8}
Multiply -4 times 8.
v=\frac{-18±\sqrt{324+352}}{2\times 8}
Multiply -32 times -11.
v=\frac{-18±\sqrt{676}}{2\times 8}
Add 324 to 352.
v=\frac{-18±26}{2\times 8}
Take the square root of 676.
v=\frac{-18±26}{16}
Multiply 2 times 8.
v=\frac{8}{16}
Now solve the equation v=\frac{-18±26}{16} when ± is plus. Add -18 to 26.
v=\frac{1}{2}
Reduce the fraction \frac{8}{16} to lowest terms by extracting and canceling out 8.
v=-\frac{44}{16}
Now solve the equation v=\frac{-18±26}{16} when ± is minus. Subtract 26 from -18.
v=-\frac{11}{4}
Reduce the fraction \frac{-44}{16} to lowest terms by extracting and canceling out 4.
8v^{2}+18v-11=8\left(v-\frac{1}{2}\right)\left(v-\left(-\frac{11}{4}\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2} for x_{1} and -\frac{11}{4} for x_{2}.
8v^{2}+18v-11=8\left(v-\frac{1}{2}\right)\left(v+\frac{11}{4}\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
8v^{2}+18v-11=8\times \frac{2v-1}{2}\left(v+\frac{11}{4}\right)
Subtract \frac{1}{2} from v by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
8v^{2}+18v-11=8\times \frac{2v-1}{2}\times \frac{4v+11}{4}
Add \frac{11}{4} to v by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
8v^{2}+18v-11=8\times \frac{\left(2v-1\right)\left(4v+11\right)}{2\times 4}
Multiply \frac{2v-1}{2} times \frac{4v+11}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
8v^{2}+18v-11=8\times \frac{\left(2v-1\right)\left(4v+11\right)}{8}
Multiply 2 times 4.
8v^{2}+18v-11=\left(2v-1\right)\left(4v+11\right)
Cancel out 8, the greatest common factor in 8 and 8.
x ^ 2 +\frac{9}{4}x -\frac{11}{8} = 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 8
r + s = -\frac{9}{4} rs = -\frac{11}{8}
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{9}{8} - u s = -\frac{9}{8} + u
Two numbers r and s sum up to -\frac{9}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{9}{4} = -\frac{9}{8}. 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{9}{8} - u) (-\frac{9}{8} + u) = -\frac{11}{8}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{11}{8}
\frac{81}{64} - u^2 = -\frac{11}{8}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{11}{8}-\frac{81}{64} = -\frac{169}{64}
Simplify the expression by subtracting \frac{81}{64} on both sides
u^2 = \frac{169}{64} u = \pm\sqrt{\frac{169}{64}} = \pm \frac{13}{8}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{9}{8} - \frac{13}{8} = -2.750 s = -\frac{9}{8} + \frac{13}{8} = 0.500
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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