Question

5. Use the elimination method to solve the linear-quadratic system y=x^2+x-42 & y=-x+6
show your work.

Answers

GPT-4o mini
To solve the linear-quadratic system of equations using the elimination method, we'll start with the two equations you provided:

1. \( y = x^2 + x - 42 \) (Quadratic equation)
2. \( y = -x + 6 \) (Linear equation)

Since both equations are equal to \( y \), we can set them equal to each other:

\[
x^2 + x - 42 = -x + 6
\]

Now, we will rearrange this equation to form a standard quadratic equation:

1. Add \( x \) and subtract \( 6 \) from both sides:

\[
x^2 + x + x - 42 - 6 = 0
\]

This simplifies to:

\[
x^2 + 2x - 48 = 0
\]

Next, we can factor the quadratic equation \( x^2 + 2x - 48 \):

To factor, we need two numbers that multiply to \(-48\) and add up to \(2\). The numbers \(8\) and \(-6\) work because:

\[
8 \cdot -6 = -48 \quad \text{and} \quad 8 + (-6) = 2
\]

Thus, we can write the factored form:

\[
(x + 8)(x - 6) = 0
\]

Setting each factor to zero gives us the possible values for \( x \):

1. \( x + 8 = 0 \) → \( x = -8 \)
2. \( x - 6 = 0 \) → \( x = 6 \)

Now, we have two values for \( x \). Next, we will substitute each of these values back into the linear equation \( y = -x + 6 \) to find the corresponding \( y \) values.

**For \( x = -8 \):**
\[
y = -(-8) + 6 = 8 + 6 = 14
\]
So, one solution is \( (-8, 14) \).

**For \( x = 6 \):**
\[
y = -(6) + 6 = -6 + 6 = 0
\]
So, another solution is \( (6, 0) \).

### Final Solutions
The solutions to the linear-quadratic system are:

\[
(-8, 14) \quad \text{and} \quad (6, 0)
\]
Shorter please
GPT-4o mini
Certainly! Here's a shorter version:

We have the system:

1. \( y = x^2 + x - 42 \)
2. \( y = -x + 6 \)

Set them equal:

\[
x^2 + x - 42 = -x + 6
\]

Rearranging gives:

\[
x^2 + 2x - 48 = 0
\]

Factoring:

\[
(x + 8)(x - 6) = 0
\]

Thus, the solutions for \( x \) are:

- \( x = -8 \)
- \( x = 6 \)

Now, substitute back to find \( y \):

1. For \( x = -8 \):
\[ y = -(-8) + 6 = 14 \]
→ Solution: \( (-8, 14) \)

2. For \( x = 6 \):
\[ y = -(6) + 6 = 0 \]
→ Solution: \( (6, 0) \)

### Final Solutions:
\[
(-8, 14) \quad \text{and} \quad (6, 0)
\]
Shorter
GPT-4o mini
Sure! Here’s an even shorter summary:

1. Set equations equal:
\[
x^2 + x - 42 = -x + 6
\]
becomes:
\[
x^2 + 2x - 48 = 0
\]

2. Factor:
\[
(x + 8)(x - 6) = 0
\]
Solutions: \( x = -8 \) and \( x = 6 \)

3. Find \( y \):
- For \( x = -8 \): \( y = 14 \) → \( (-8, 14) \)
- For \( x = 6 \): \( y = 0 \) → \( (6, 0) \)

### Solutions:
\[
(-8, 14) \quad \text{and} \quad (6, 0)
\]