Question

To show how to solve the equation, create two functions and find their intersection points. What two functions can be used to solve the following equation, and what is their solution set?

10 − x2 = x + 4

Responses

f(x) = 10 + x2, g(x) = x + 4, {−3, 1}

g(x) = x + 4. f(x) = 10 − x^2, {1, 6}

g(x) = x + 4, f(x) = 10 + x^2, {−3, 2}

f(x) = 10 −x2, g(x) = x + 4, {−3, 2}

Answer 1

To solve the equation \(10 - x^2 = x + 4\), we can rewrite it in the form of two functions and find their intersection.

1. Define the two functions:
- \(f(x) = 10 - x^2\)
- \(g(x) = x + 4\)

2. To find the intersection points, we need to set \(f(x)\) equal to \(g(x)\):
\[
10 - x^2 = x + 4
\]

3. Rearranging the equation gives:
\[
-x^2 - x + 10 - 4 = 0
\]
\[
-x^2 - x + 6 = 0
\]

4. Multiply the entire equation by \(-1\) to simplify:
\[
x^2 + x - 6 = 0
\]

5. We can factor this quadratic equation:
\[
(x + 3)(x - 2) = 0
\]

6. Setting each factor to zero gives the solutions:
\[
x + 3 = 0 \implies x = -3
\]
\[
x - 2 = 0 \implies x = 2
\]

Thus, the solution set is \(\{-3, 2\}\).

Given the options, the correct response is:
**f(x) = 10 − x^2, g(x) = x + 4, {−3, 2}**.