Question

Help me please with this equation

Answer 1

Answer: The function has two roots in common.

Explanation:

To determine how many roots the function
`\( f(x) = x^2 - 4x - 5 \)` has, we can find its discriminant. The discriminant will tell us the nature of the roots:

1. If
`\( \Delta > 0 \)`, the quadratic has two distinct real roots.

2. If
`\( \Delta = 0 \)`, the quadratic has one real root (a repeated root).

3. If
`\( \Delta < 0 \)`, the quadratic has no real roots (two complex conjugate roots).

The discriminant
`\( \Delta \)` for a quadratic equation of the form
`\( ax^2 + bx + c \)`is given by:

`\[ \Delta = b^2 - 4ac \]`

For the function
`\( f(x) = x^2 - 4x - 5 \)`:

a = 1

b = -4

c = -5

Plugging in these values:

`\[ \Delta = (-4)^2 - 4(1)(-5) \]`

Let's calculate the value of
`\( \Delta \)` to determine the number of roots.

The discriminant
`\( \Delta \)` for the function
`\( f(x) = x^2 - 4x - 5 \)` is:

`\[ \Delta = 36 \]`

Since
`\( \Delta > 0 \)`, the quadratic function
`\( f(x) = x^2 - 4x - 5 \)` has two distinct real roots.

Thus, the function has two roots in common.