How many intercepts does `y = x^2 − 5x + 6` have?

You can find this function's `x`-intercepts by making equal to zero and its `y`-intercepts by evaluating the function for `x=0`.

For the `x`-intercepts, you have

`x^2 - 5x + 6 = 0`

You can determine how many solutions this quadratic has by calculating its discriminant, `Delta`, which, for a general form quadratic equation

`color(blue)(ax^2 + bx + c = 0)`

takes the form

`color(blue)(Delta = b^2 - 4ac)`

In your case, the discriminant will be

`Delta = (-5)^2 - 4 * 1 * 6`

`Delta = 25 - 24 = color(green)(1)`

When `Delta>0`, the quadratic equation has two distinct real roots that take the form

`color(blue)(x_(1,2) = (-b +- sqrt(Delta))/(2a))`

In your case, these roots will be

`x_(1,2) = (-(-5) +- sqrt(1))/(2 * 1)`

`x_(1,2) = (5 +- 1)/2 = {(x_1 = (5 + 1)/2 = 3), (x_2 = (5-1)/2 = 2) :}`

This means that the function will have two `x`-intercepts at `x=2` and `x=3`.

The `y`-intercept will be

`y = (0)^2 - 5 * (0) + 6 = 6`

The function will intercept the `x`-axis in the points `(2,0)` and `(3,0)`, and the `y`-axis in the point `(0, 6)`.

graph{x^2 - 5x + 6 [-10, 10, -5, 5]}