Create two functions and their intersection points what to functions can be used to solve the following equation 10 - x squared equals x + 4

Answer 1

To solve the equation \(10 - x^2 = x + 4\), we can rearrange it to form two functions whose intersection points we want to find:

1. Let \(f(x) = 10 - x^2\) (the left side of the equation)
2. Let \(g(x) = x + 4\) (the right side of the equation)

So, we want to find the intersection points of the two functions \(f(x)\) and \(g(x)\).

To find the intersection points, we need to set the two functions equal to each other:

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

Now, rearranging the equation gives:

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

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

Multiplying through by \(-1\) for simplification:

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

Next, we can factor or use the quadratic formula to solve for \(x\).

Factoring the quadratic:

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

Setting each factor to zero gives us the solutions:

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

Thus, the \(x\)-coordinates of the intersection points are \(x = -3\) and \(x = 2\).

To find the corresponding \(y\)-coordinates for both intersection points, we can substitute \(x\) back into either \(f(x)\) or \(g(x)\):

1. For \(x = -3\): \[ f(-3) = 10 - (-3)^2 = 10 - 9 = 1 \] So, the point is \((-3, 1)\).

2. For \(x = 2\): \[ f(2) = 10 - (2)^2 = 10 - 4 = 6 \] So, the point is \((2, 6)\).

Intersection Points

The intersection points of the two functions \(f(x) = 10 - x^2\) and \(g(x) = x + 4\) are:

1. \((-3, 1)\)
2. \((2, 6)\)