Question

Factor 3x^2 - 13x - 10

Answer 1

To factor the quadratic expression \( 3x^2 - 13x - 10 \), we can look for two numbers that multiply to \( a \cdot c \) (where \( a = 3 \) and \( c = -10 \)) and add up to \( b \) (where \( b = -13 \)).

1. Calculate \( a \cdot c \):
\[
a \cdot c = 3 \cdot (-10) = -30
\]

2. We need to find two numbers that multiply to \(-30\) and add to \(-13\). The numbers that work are \(-15\) and \(2\):
\[
-15 \cdot 2 = -30 \quad \text{and} \quad -15 + 2 = -13
\]

3. Rewrite the middle term \(-13x\) using \(-15x\) and \(2x\):
\[
3x^2 - 15x + 2x - 10
\]

4. Now, group the terms:
\[
(3x^2 - 15x) + (2x - 10)
\]

5. Factor out the common terms in each group:
\[
3x(x - 5) + 2(x - 5)
\]

6. Now factor out the common binomial factor \((x - 5)\):
\[
(3x + 2)(x - 5)
\]

Thus, the factored form of \( 3x^2 - 13x - 10 \) is:
\[
(3x + 2)(x - 5)
\]