Question

Order the steps from first to last that would be used to factor:

3x^2 − 13x − 10

(7 points)

Answer 1

To factor the quadratic expression \(3x^2 - 13x - 10\), you can follow these steps in order:

1. **Identify** the coefficients: For the quadratic \(ax^2 + bx + c\), here \(a = 3\), \(b = -13\), and \(c = -10\).

2. **Multiply** \(a\) and \(c\): Calculate \(3 \times -10 = -30\).

3. **Find two numbers** that multiply to \(-30\) (the product from step 2) and add to \(-13\) (the coefficient \(b\)). The two numbers are \(-15\) and \(2\) because \(-15 + 2 = -13\) and \(-15 \times 2 = -30\).

4. **Rewrite** the middle term: Replace \(-13x\) with \(-15x + 2x\):
\[
3x^2 - 15x + 2x - 10
\]

5. **Group** the terms: Group the first two terms and the last two terms:
\[
(3x^2 - 15x) + (2x - 10)
\]

6. **Factor out** the common factors from each group:
\[
3x(x - 5) + 2(x - 5)
\]

7. **Factor out the common binomial** \((x - 5)\):
\[
(3x + 2)(x - 5)
\]

So, the expression \(3x^2 - 13x - 10\) factors to \((3x + 2)(x - 5)\).

The ordered steps are as follows:
1. Identify the coefficients.
2. Multiply \(a\) and \(c\).
3. Find two numbers that multiply to \(-30\) and add to \(-13\).
4. Rewrite the middle term.
5. Group the terms.
6. Factor out the common factors from each group.
7. Factor out the common binomial.