Question

To show how to solve the equation by graphing, 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

Desmos Link

(1 point)
Responses

g(x) = x + 4, f(x) = 10+x2, {−3,2}
g(x) = x + 4, f(x) = 10+x2, {−3,2}

g(x) = x + 4, f(x) = 10+x2, {−3,1}
g(x) = x + 4, f(x) = 10+x2, {−3,1}

g(x) = x + 4, f(x) = 10−x2, {1,6}
g(x) = x + 4, f(x) = 10−x2, {1,6}

g(x) = x + 4, f(x) = 10−x2, {−3,2}

Answer 1

To solve the equation \( 10 - x^2 = x + 4 \) by graphing, we can rewrite this equation as two separate functions:

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

Now, we want to find the intersection points of these two functions, which are the solutions to the equation \( 10 - x^2 = x + 4 \).

To find the points of intersection, we set \( g(x) = f(x) \):

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

Rearranging this gives:

\[
-x^2 - x + 10 - 4 = 0
\]
\[
-x^2 - x + 6 = 0
\]

Multiplying through by -1 gives:

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

Now, we can factor this quadratic equation:

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

Setting each factor to zero gives the solutions:

\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]

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

Based on the provided options, the correct response is:

**g(x) = x + 4, f(x) = 10 − x², {−3, 2}**