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?

Question

10 - x^2 = x+4
Answers:

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

g(x) = x + 4, f(x) =10 - x^2, {1,6}

f(x) = 10 + x^2, 9 (x) =x +4,{-3,1}

f(x) =10 - x^2, 9 (x) =x+4,{-3,2}

Answer 1

To solve the equation \( 10 - x^2 = x + 4 \) by creating two functions and finding their intersection points, we can rearrange the equation to identify \( f(x) \) and \( g(x) \):

1. Rewrite the equation in function form:
   • Let \( f(x) = 10 - x^2 \)
   • Let \( g(x) = x + 4 \)

Now, we want to find the intersection points of the functions \( f(x) \) and \( g(x) \), which means we need to solve the equation:

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

2. Set the two functions equal to each other:

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

3. Rearrange the equation:

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

This simplifies to:

\[ 6 = x + x^2 \]

Rearranging further gives us:

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

4. Now we can factor this quadratic equation:

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

Setting each factor to zero gives:

\[ x - 2 = 0 \quad \text{or} \quad x + 3 = 0 \] \[ x = 2 \quad \text{or} \quad x = -3 \]

5. Thus, the solution set for the equation \( 10 - x^2 = x + 4 \) is:

\[ {-3, 2} \]

From the options provided, the correct pair of functions and solution are:

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