Step 1: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:
Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -9} $.
$$ a \cdot c = -18 $$Step 3: Find out two numbers that multiply to $ a \cdot c = -18 $ and add to $ b = 3 $.
Step 4: All pairs of numbers with a product of $ -18 $ are:
PRODUCT = -18 | |
-1 18 | 1 -18 |
-2 9 | 2 -9 |
-3 6 | 3 -6 |
Step 5: Find out which factor pair sums up to $\color{blue}{ b = 3 }$
PRODUCT = -18 and SUM = 3 | |
-1 18 | 1 -18 |
-2 9 | 2 -9 |
-3 6 | 3 -6 |
Step 6: Replace middle term $ 3 x $ with $ 6x-3x $:
$$ 2x^{2}+3x-9 = 2x^{2}+6x-3x-9 $$Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -3 $ out of the last two terms.
$$ 2x^{2}+6x-3x-9 = 2x\left(x+3\right) -3\left(x+3\right) = \left(2x-3\right) \left(x+3\right) $$