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.
- \( f(x) = 10 - x^2 \)
- \( g(x) = x + 4 \)
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:
- \(x + 3 = 0 \Rightarrow x = -3\)
- \(x - 2 = 0 \Rightarrow x = 2\)
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 \({-3, 2}\).