Factor
\left(a-11\right)\left(a-2\right)
Evaluate
\left(a-11\right)\left(a-2\right)
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p+q=-13 pq=1\times 22=22
Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa+22. To find p and q, set up a system to be solved.
-1,-22 -2,-11
Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. List all such integer pairs that give product 22.
-1-22=-23 -2-11=-13
Calculate the sum for each pair.
p=-11 q=-2
The solution is the pair that gives sum -13.
\left(a^{2}-11a\right)+\left(-2a+22\right)
Rewrite a^{2}-13a+22 as \left(a^{2}-11a\right)+\left(-2a+22\right).
a\left(a-11\right)-2\left(a-11\right)
Factor out a in the first and -2 in the second group.
\left(a-11\right)\left(a-2\right)
Factor out common term a-11 by using distributive property.
a^{2}-13a+22=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.
a=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 22}}{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.
a=\frac{-\left(-13\right)±\sqrt{169-4\times 22}}{2}
Square -13.
a=\frac{-\left(-13\right)±\sqrt{169-88}}{2}
Multiply -4 times 22.
a=\frac{-\left(-13\right)±\sqrt{81}}{2}
Add 169 to -88.
a=\frac{-\left(-13\right)±9}{2}
Take the square root of 81.
a=\frac{13±9}{2}
The opposite of -13 is 13.
a=\frac{22}{2}
Now solve the equation a=\frac{13±9}{2} when ± is plus. Add 13 to 9.
a=11
Divide 22 by 2.
a=\frac{4}{2}
Now solve the equation a=\frac{13±9}{2} when ± is minus. Subtract 9 from 13.
a=2
Divide 4 by 2.
a^{2}-13a+22=\left(a-11\right)\left(a-2\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 11 for x_{1} and 2 for x_{2}.
x ^ 2 -13x +22 = 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 = 13 rs = 22
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{13}{2} - u s = \frac{13}{2} + u
Two numbers r and s sum up to 13 exactly when the average of the two numbers is \frac{1}{2}*13 = \frac{13}{2}. 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{13}{2} - u) (\frac{13}{2} + u) = 22
To solve for unknown quantity u, substitute these in the product equation rs = 22
\frac{169}{4} - u^2 = 22
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = 22-\frac{169}{4} = -\frac{81}{4}
Simplify the expression by subtracting \frac{169}{4} on both sides
u^2 = \frac{81}{4} u = \pm\sqrt{\frac{81}{4}} = \pm \frac{9}{2}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =\frac{13}{2} - \frac{9}{2} = 2 s = \frac{13}{2} + \frac{9}{2} = 11
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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