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−x^2=x+4

Answer 1

To solve the equation \(10 - x^2 = x + 4\), we can rearrange it into a standard form suitable for analysis by creating two functions.

First, let's rearrange the equation to one side:

\[
10 - x^2 - x - 4 = 0
\]

Simplifying this, we get:

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

Multiplying through by -1 for convenience gives us:

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

### Creating the Functions

We will define two functions based on the left-hand and right-hand sides of the original equation before rearranging.

1. \( f(x) = 10 - x^2 \)
2. \( g(x) = x + 4 \)

### Finding the Intersection Points

To find the intersection points of these two functions, we need to find the values of \(x\) where \(f(x) = g(x)\):

\[
10 - x^2 = x + 4
\]

Setting these two equal gives us:

\[
10 - x^2 - x - 4 = 0
\]

This simplifies to:

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

We already rearranged it into:

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

### Solving the Quadratic Equation

To solve the quadratic equation \(x^2 + x - 6 = 0\), we can factor it:

\[
(x + 3)(x - 2) = 0
\]

Setting each factor to zero gives us:

1. \(x + 3 = 0 \Rightarrow x = -3\)
2. \(x - 2 = 0 \Rightarrow x = 2\)

### Solution Set

Thus, the solutions to the equation \(10 - x^2 = x + 4\) are:

\[
x = -3 \quad \text{and} \quad x = 2
\]

The solution set is:

\[
\{-3, 2\}
\]

### Conclusion

To summarize, the two functions created are \( f(x) = 10 - x^2 \) and \( g(x) = x + 4 \). Their intersection points, which solve the original equation, are \(x = -3\) and \(x = 2\). Therefore, the solution set is \(\{-3, 2\}\).