Factor
\left(x+11\right)\left(8x+1\right)
Evaluate
\left(x+11\right)\left(8x+1\right)
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a+b=89 ab=8\times 11=88
Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx+11. To find a and b, set up a system to be solved.
1,88 2,44 4,22 8,11
Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 88.
1+88=89 2+44=46 4+22=26 8+11=19
Calculate the sum for each pair.
a=1 b=88
The solution is the pair that gives sum 89.
\left(8x^{2}+x\right)+\left(88x+11\right)
Rewrite 8x^{2}+89x+11 as \left(8x^{2}+x\right)+\left(88x+11\right).
x\left(8x+1\right)+11\left(8x+1\right)
Factor out x in the first and 11 in the second group.
\left(8x+1\right)\left(x+11\right)
Factor out common term 8x+1 by using distributive property.
8x^{2}+89x+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.
x=\frac{-89±\sqrt{89^{2}-4\times 8\times 11}}{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.
x=\frac{-89±\sqrt{7921-4\times 8\times 11}}{2\times 8}
Square 89.
x=\frac{-89±\sqrt{7921-32\times 11}}{2\times 8}
Multiply -4 times 8.
x=\frac{-89±\sqrt{7921-352}}{2\times 8}
Multiply -32 times 11.
x=\frac{-89±\sqrt{7569}}{2\times 8}
Add 7921 to -352.
x=\frac{-89±87}{2\times 8}
Take the square root of 7569.
x=\frac{-89±87}{16}
Multiply 2 times 8.
x=-\frac{2}{16}
Now solve the equation x=\frac{-89±87}{16} when ± is plus. Add -89 to 87.
x=-\frac{1}{8}
Reduce the fraction \frac{-2}{16} to lowest terms by extracting and canceling out 2.
x=-\frac{176}{16}
Now solve the equation x=\frac{-89±87}{16} when ± is minus. Subtract 87 from -89.
x=-11
Divide -176 by 16.
8x^{2}+89x+11=8\left(x-\left(-\frac{1}{8}\right)\right)\left(x-\left(-11\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}{8} for x_{1} and -11 for x_{2}.
8x^{2}+89x+11=8\left(x+\frac{1}{8}\right)\left(x+11\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
8x^{2}+89x+11=8\times \frac{8x+1}{8}\left(x+11\right)
Add \frac{1}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
8x^{2}+89x+11=\left(8x+1\right)\left(x+11\right)
Cancel out 8, the greatest common factor in 8 and 8.
x ^ 2 +\frac{89}{8}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{89}{8} 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{89}{16} - u s = -\frac{89}{16} + u
Two numbers r and s sum up to -\frac{89}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{89}{8} = -\frac{89}{16}. 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{89}{16} - u) (-\frac{89}{16} + u) = \frac{11}{8}
To solve for unknown quantity u, substitute these in the product equation rs = \frac{11}{8}
\frac{7921}{256} - u^2 = \frac{11}{8}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = \frac{11}{8}-\frac{7921}{256} = -\frac{7569}{256}
Simplify the expression by subtracting \frac{7921}{256} on both sides
u^2 = \frac{7569}{256} u = \pm\sqrt{\frac{7569}{256}} = \pm \frac{87}{16}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{89}{16} - \frac{87}{16} = -11 s = -\frac{89}{16} + \frac{87}{16} = -0.125
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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