Question

Which expression is equivalent to the trinomial?
Ex) 2x^2 + 5x - 3
2) 3x^2 + 22x + 7
type your answers to each problem in the table below. The order does not matter.

Answer 1

To factor the given trinomials, we will follow the steps of finding two numbers that multiply to \( a \cdot c \) (the product of the coefficient of \( x^2 \) and the constant term) and add to \( b \) (the coefficient of \( x \)).

Let's analyze both examples:

1. **For \( 2x^2 + 5x - 3 \)**:
- Here, \( a = 2 \), \( b = 5 \), \( c = -3 \).
- \( a \cdot c = 2 \cdot -3 = -6 \).
- We need two numbers that multiply to \(-6\) and add to \(5\).
- The two numbers are \(6\) and \(-1\).
- Rewrite the expression as: \( 2x^2 + 6x - 1x - 3 \).
- Factor by grouping:
- \( 2x(x + 3) - 1(x + 3) \)
- \( (2x - 1)(x + 3) \)
- The equivalent expression for \( 2x^2 + 5x - 3 \) is \( (2x - 1)(x + 3) \).

2. **For \( 3x^2 + 22x + 7 \)**:
- Here, \( a = 3 \), \( b = 22 \), \( c = 7 \).
- \( a \cdot c = 3 \cdot 7 = 21 \).
- We need two numbers that multiply to \( 21 \) and add to \( 22 \).
- The two numbers are \(21\) and \(1\).
- Rewrite the expression as: \( 3x^2 + 21x + 1x + 7 \).
- Factor by grouping:
- \( 3x(x + 7) + 1(x + 7) \)
- \( (3x + 1)(x + 7) \)
- The equivalent expression for \( 3x^2 + 22x + 7 \) is \( (3x + 1)(x + 7) \).

### Summary of Equivalent Expressions:
- For \( 2x^2 + 5x - 3 \): **\( (2x - 1)(x + 3) \)**
- For \( 3x^2 + 22x + 7 \): **\( (3x + 1)(x + 7) \)**

Feel free to provide any other trinomial expressions for further assistance!