Decide equation using only factors to solve this problem 12x^{2} + 5x - 2 = 0

Question

Decide equation using only factors to solve this problem

12 x^{2} + 5 x - 2 = 0

Answer 1

In order to factor the quadratic, we need to find the values of a, b and c for the given quadratic $12 x^{2} + 5 x - 2 = 0$ on comparing with standard quadratic
$a x^{2} + b x + c = 0$
$12 x^{2} + 5 x - 2 = 0$
$a = 12 b = 5, c = - 2.$
Now, let us apply ac method to factor the quadratic.
We got $a c = - 24 a n d b = 5.$ So, we need to find the factors of -24, those add upto 5.
$a c = 12 \cdot - 2 = - 24$ (Product)
$b = 5$ (Sum)
-24 could be factored in two numbers 8 and -3 that add upto 5.
Therefore, middle term 5x of expression $12 x^{2} + 5 x - 2 = 0,$ could be break into two terms 8x and -3x.
Then we need to make the break the expression into groups.
Factor out 3x form $12 x^{2} - 3 x$ and factor out 2 from:
$8 x - 2.$
Then factor out common parenthesis $(4 x - 1) .$
We got factored form $(4 x - 1) (3 x + 2) = 0$
$12 x^{2} + 5 x - 2 = 0$
$(12 x^{2} - 3 x) + (8 x - 2) = 0$
$3 x (4 x - 1) + 2 (4 x - 1) = 0$
Factor out common parenthesis $(4 x - 1)$
$(4 x - 1) (3 x + 2) = 0$
By applying zero product rule, we need to set each factor equal to zero and solve for x.
We got $x = \frac{1}{4}, \frac{- 2}{3} .$
$4 x - 1 = 0 o r 3 x + 2 = 0$
$+ 1 + 1 - 2 - 2$
$\frac{4 x}{4} = \frac{1}{4} o r \frac{3 x}{3} = \frac{- 2}{3}$
$x = 1.4 o r x = \frac{- 2}{3}$

Decide equation using only factors to solve this problem 12x^{2} + 5x - 2 = 0

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