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p+q=22 pq=3\left(-16\right)=-48
Factor the expression by grouping. First, the expression needs to be rewritten as 3b^{2}+pb+qb-16. To find p and q, set up a system to be solved.
-1,48 -2,24 -3,16 -4,12 -6,8
Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -48.
-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2
Calculate the sum for each pair.
p=-2 q=24
The solution is the pair that gives sum 22.
\left(3b^{2}-2b\right)+\left(24b-16\right)
Rewrite 3b^{2}+22b-16 as \left(3b^{2}-2b\right)+\left(24b-16\right).
b\left(3b-2\right)+8\left(3b-2\right)
Factor out b in the first and 8 in the second group.
\left(3b-2\right)\left(b+8\right)
Factor out common term 3b-2 by using distributive property.
3b^{2}+22b-16=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{-22±\sqrt{22^{2}-4\times 3\left(-16\right)}}{2\times 3}
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{-22±\sqrt{484-4\times 3\left(-16\right)}}{2\times 3}
Square 22.
b=\frac{-22±\sqrt{484-12\left(-16\right)}}{2\times 3}
Multiply -4 times 3.
b=\frac{-22±\sqrt{484+192}}{2\times 3}
Multiply -12 times -16.
b=\frac{-22±\sqrt{676}}{2\times 3}
Add 484 to 192.
b=\frac{-22±26}{2\times 3}
Take the square root of 676.
b=\frac{-22±26}{6}
Multiply 2 times 3.
b=\frac{4}{6}
Now solve the equation b=\frac{-22±26}{6} when ± is plus. Add -22 to 26.
b=\frac{2}{3}
Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.
b=-\frac{48}{6}
Now solve the equation b=\frac{-22±26}{6} when ± is minus. Subtract 26 from -22.
b=-8
Divide -48 by 6.
3b^{2}+22b-16=3\left(b-\frac{2}{3}\right)\left(b-\left(-8\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{3} for x_{1} and -8 for x_{2}.
3b^{2}+22b-16=3\left(b-\frac{2}{3}\right)\left(b+8\right)
Simplify all the expressions of the form p-\left(-q\right) to p+q.
3b^{2}+22b-16=3\times \frac{3b-2}{3}\left(b+8\right)
Subtract \frac{2}{3} from b by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
3b^{2}+22b-16=\left(3b-2\right)\left(b+8\right)
Cancel out 3, the greatest common factor in 3 and 3.
x ^ 2 +\frac{22}{3}x -\frac{16}{3} = 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 3
r + s = -\frac{22}{3} rs = -\frac{16}{3}
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{11}{3} - u s = -\frac{11}{3} + u
Two numbers r and s sum up to -\frac{22}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{22}{3} = -\frac{11}{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>
(-\frac{11}{3} - u) (-\frac{11}{3} + u) = -\frac{16}{3}
To solve for unknown quantity u, substitute these in the product equation rs = -\frac{16}{3}
\frac{121}{9} - u^2 = -\frac{16}{3}
Simplify by expanding (a -b) (a + b) = a^2 – b^2
-u^2 = -\frac{16}{3}-\frac{121}{9} = -\frac{169}{9}
Simplify the expression by subtracting \frac{121}{9} on both sides
u^2 = \frac{169}{9} u = \pm\sqrt{\frac{169}{9}} = \pm \frac{13}{3}
Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u
r =-\frac{11}{3} - \frac{13}{3} = -8 s = -\frac{11}{3} + \frac{13}{3} = 0.667
The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.