Question

0=6x2+13x+5 .Write quadratic equation as the product of two factors

Answer 1

To factor the quadratic equation \( 6x^2 + 13x + 5 \), we want to express it in the form of \( (ax + b)(cx + d) \).

1. **Identify coefficients**: For the quadratic equation \( ax^2 + bx + c \):
- \( a = 6 \)
- \( b = 13 \)
- \( c = 5 \)

2. **Multiply \( a \) and \( c \)**: \( 6 \times 5 = 30 \).

3. **Find two numbers that multiply to 30 and add up to 13**:
- The two numbers are \( 3 \) and \( 10 \) (since \( 3 \times 10 = 30 \) and \( 3 + 10 = 13 \)).

4. **Rewrite the middle term**:
\[
6x^2 + 3x + 10x + 5
\]

5. **Group the terms**:
\[
(6x^2 + 3x) + (10x + 5)
\]

6. **Factor by grouping**:
- From the first group \( 6x^2 + 3x \), factor out \( 3x \):
\[
3x(2x + 1)
\]
- From the second group \( 10x + 5 \), factor out \( 5 \):
\[
5(2x + 1)
\]

7. **Combine the factored expressions**:
\[
3x(2x + 1) + 5(2x + 1) = (2x + 1)(3x + 5)
\]

The quadratic equation can thus be expressed as the product of two factors:
\[
6x^2 + 13x + 5 = (2x + 1)(3x + 5)
\]