Factorize: 12x² + 25x − 22

SOLUTION:

We equate our equation to the generic form of a quadratic equation: \[ \\ \\ \] \[ 12x^{2} + 25x - 22 = ax^{2} + bx + c \] \[ \\ \\ \] We compare coefficients a, b, and c with the corresponding coefficients on the LHS: \[ a = 12 \;;\; b = 25 \;;\; c = -22 \] \[ \\ \\ \\ \] We apply the ac–b technique to factor the quadratic. We look for two numbers whose product is ac and whose sum is b. \[ ac = -264 \;;\; b = 25 \quad\Rightarrow\quad [44 \times (-6)] \; ; \; [33 \times (-8)] \] \[ \\ \\ \\ \] From the two factors of ac found, it is already seen that the sum of the factors 33 and -8 would produce 25, which is b. Hence: \[ \\ \] \[ \begin{aligned} 12x^{2} + 25x - 22 &= 12x^{2} + 33x - 8x - 22 \\ \\ &= 3x(4x + 11) - 2(4x + 11) \\ \\ \therefore \; 12x^{2} + 25x - 22 &= \underline{\underline{(3x - 2)(4x + 11)}} \end{aligned} \] \[ \\ \] and the equation is thus factorized.