How do I write the equation 3x² + 19x + 14 in standard form and factor the left side?

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)

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