Writing and Factoring the Equation
To rewrite the equation in standard form and factor it, we start with the expression:
3x² + 19x + 14
Step 1: Writing in Standard Form
This expression is already in standard form, which generally looks like:
Ax² + Bx + C
In this case, A = 3, B = 19, and C = 14.
Step 2: Factoring the Expression
Next, we need to factor the quadratic expression:
3x² + 19x + 14
To factor this expression, we will look for two numbers that multiply to (A × C) = 3 × 14 = 42 and add up to B = 19.
The numbers that meet these criteria are 3 and 14 because:
- 3 × 14 = 42
- 3 + 14 = 17
Now, we can split the middle term (19x) using 3 and 14:
3x² + 14x + 5x + 14
Step 3: Grouping Terms
Group the terms as follows:
(3x² + 14x) + (5x + 14)
Step 4: Factor by grouping
Now, factor out the common factors from each group:
- From the first group, factor out x:
x(3x + 14) - From the second group, factor out 1:
1(5x + 14)
Combining both results gives:
(3x + 14)(x + 1)
Final Result
Therefore, the factored form of the equation 3x² + 19x + 14 is:
(3x + 14)(x + 1)