Question
Which method would be best to solve n^2 - n - 6 = 0
Why?
Why?
Answers
Answered by
GPT-4o mini
To solve the quadratic equation \( n^2 - n - 6 = 0 \), the most suitable method would be to factor the equation. This is because the equation is relatively simple and can be factored into two binomials.
Here's how you can do it:
1. **Rearrange the equation:** It is already in standard form, \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -1 \), and \( c = -6 \).
2. **Look for two numbers that multiply to \( ac \) (which is \( 1 \cdot -6 = -6 \)) and add to \( b \) (which is \(-1\)).** The numbers that fit these criteria are \( -3 \) and \( 2 \) because:
- \( -3 \times 2 = -6 \)
- \( -3 + 2 = -1 \)
3. **Factor the quadratic:**
\[
n^2 - n - 6 = (n - 3)(n + 2) = 0
\]
4. **Set each factor to zero:**
\[
n - 3 = 0 \quad \Rightarrow \quad n = 3
\]
\[
n + 2 = 0 \quad \Rightarrow \quad n = -2
\]
Thus, the solutions to the equation \( n^2 - n - 6 = 0 \) are \( n = 3 \) and \( n = -2 \).
**Why use factoring?** Factoring is a quick and efficient method for solving quadratic equations when they have rational roots and can be expressed in a simple binomial form. In this case, it avoids the need for using the quadratic formula, which is more complex and unnecessary for a straightforward quadratic like this one.
Here's how you can do it:
1. **Rearrange the equation:** It is already in standard form, \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -1 \), and \( c = -6 \).
2. **Look for two numbers that multiply to \( ac \) (which is \( 1 \cdot -6 = -6 \)) and add to \( b \) (which is \(-1\)).** The numbers that fit these criteria are \( -3 \) and \( 2 \) because:
- \( -3 \times 2 = -6 \)
- \( -3 + 2 = -1 \)
3. **Factor the quadratic:**
\[
n^2 - n - 6 = (n - 3)(n + 2) = 0
\]
4. **Set each factor to zero:**
\[
n - 3 = 0 \quad \Rightarrow \quad n = 3
\]
\[
n + 2 = 0 \quad \Rightarrow \quad n = -2
\]
Thus, the solutions to the equation \( n^2 - n - 6 = 0 \) are \( n = 3 \) and \( n = -2 \).
**Why use factoring?** Factoring is a quick and efficient method for solving quadratic equations when they have rational roots and can be expressed in a simple binomial form. In this case, it avoids the need for using the quadratic formula, which is more complex and unnecessary for a straightforward quadratic like this one.