What is the midpoint of the x intercepts of f(x) = x^2 + x – 4?

To find the midpoint of the x intercepts of the function f(x) = x² + x – 4, we first need to determine the x intercepts themselves. The x intercepts occur where the function equals zero, so we set the equation to zero:

x² + x – 4 = 0

Next, we can use the quadratic formula to find the roots of this equation, which is given by:

x = (-b ± √(b² – 4ac)) / (2a)

In our case, a = 1, b = 1, and c = -4.

Substituting these values into the formula, we get:

x = ( -1 ± √(1² – 4(1)(-4)) ) / (2(1))

This simplifies to:

x = ( -1 ± √(1 + 16) ) / 2

x = ( -1 ± √17 ) / 2

This means we have two x intercepts:

x₁ = ( -1 + √17 ) / 2

x₂ = ( -1 – √17 ) / 2

To find the midpoint of these two x intercepts, we use the midpoint formula:

Midpoint = (x₁ + x₂) / 2

Now, substituting in our x intercepts:

Midpoint = [ ( -1 + √17 ) / 2 + ( -1 – √17 ) / 2 ] / 2

This simplifies to:

Midpoint = ( -1 + √17 – 1 – √17 ) / 4

Midpoint = -2 / 4

Midpoint = -1/2

Therefore, the midpoint of the x intercepts of the function f(x) = x² + x – 4 is -1/2.