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p+q=-6 pq=1\left(-91\right)=-91
Factor the expression by grouping. First, the expression needs to be rewritten as b^{2}+pb+qb-91. To find p and q, set up a system to be solved.
1,-91 7,-13
Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -91.
1-91=-90 7-13=-6
Calculate the sum for each pair.
p=-13 q=7
The solution is the pair that gives sum -6.
\left(b^{2}-13b\right)+\left(7b-91\right)
Rewrite b^{2}-6b-91 as \left(b^{2}-13b\right)+\left(7b-91\right).
b\left(b-13\right)+7\left(b-13\right)
Factor out b in the first and 7 in the second group.
\left(b-13\right)\left(b+7\right)
Factor out common term b-13 by using distributive property.
b^{2}-6b-91=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.
b=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-91\right)}}{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.
b=\frac{-\left(-6\right)±\sqrt{36-4\left(-91\right)}}{2}
Square -6.
b=\frac{-\left(-6\right)±\sqrt{36+364}}{2}
Multiply -4 times -91.
b=\frac{-\left(-6\right)±\sqrt{400}}{2}
Add 36 to 364.
b=\frac{-\left(-6\right)±20}{2}
Take the square root of 400.
b=\frac{6±20}{2}
The opposite of -6 is 6.
b=\frac{26}{2}
Now solve the equation b=\frac{6±20}{2} when ± is plus. Add 6 to 20.
b=13
Divide 26 by 2.
b=-\frac{14}{2}
Now solve the equation b=\frac{6±20}{2} when ± is minus. Subtract 20 from 6.
b=-7
Divide -14 by 2.
b^{2}-6b-91=\left(b-13\right)\left(b-\left(-7\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 13 for x_{1} and -7 for x_{2}.
b^{2}-6b-91=\left(b-13\right)\left(b+7\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
x ^ 2 -6x -91 = 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 = 6 rs = -91
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 = 3 - u s = 3 + u
Two numbers r and s sum up to 6 exactly when the average of the two numbers is \frac{1}{2}*6 = 3. 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>
(3 - u) (3 + u) = -91
To solve for unknown quantity u, substitute these in the product equation rs = -91
9 - u^2 = -91
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -91-9 = -100
Simplify the expression by subtracting 9 on both sides
u^2 = 100 u = \pm\sqrt{100} = \pm 10
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =3 - 10 = -7 s = 3 + 10 = 13
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.