To write the equation in standard form, we need to ensure it is arranged as ax² + bx + c = 0. The given equation is:
3x² + 19x + 14 = 0
Here, the equation is already in standard form where a = 3, b = 19, and c = 14.
Next, we will factor the left side of the equation. We are looking for two numbers that multiply to a * c = 3 * 14 = 42 and add up to b = 19.
The numbers that satisfy these conditions are 14 and 3, as:
- 14 * 3 = 42
- 14 + 3 = 17
So we rewrite the middle term (19x) using 14 and 3:
3x² + 14x + 5x + 14 = 0
Next, we can factor by grouping:
Group the first two terms and the last two terms:
(3x² + 14x) + (5x + 14) = 0
Factor out the common factors:
x(3x + 14) + 1(5x + 14) = 0
Now we can factor further:
=(3x + 2)(x + 7) = 0
So, the left side of the equation factors to:
(3x + 2)(x + 7) = 0