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Can all quadratic functions be written in intercept form?

The simple answer is: No, not all quadratic functions can be written in the intercept form. Only those quadratic functions can be written in the intercept form that have real solutions.

Let us understand this further.

What is intercept form?

The intercept form of a quadratic equation is as follows:
`y = a(x - p)(x - q)`
where 'p' and 'q' are the roots of the quadratic function. We know that a quadratic function has 'roots' or 'zeros' only when its graph touches or intersects the x-axis.

Functions that can be written in intercept form

Some quadratic functions have real solutions; That is, their graphs touch or intersect the x-axis at some point/s. There graphs look like either of the two graphs given below:

Graph touches the x-axis. This type of quadratic function has only one root and its intercept form is of the form `y = a(x - p)^2`

Graph intersects the x-axis at two points, say at x = p and x = q; This type of quadratic function has intercept form `y = a(x - p)(x - q)`
Such quadratic functions, that have a real solution, can be written in intercept form. This is because the roots or zeros (or real solutions) of the quadratic function are the values of 'p' and 'q' in the intercept form.

Functions that can't be written in intercept form

On the other hand,some quadratic functions do not have any real solution. Their graph does not touch or intersect the x-axis at any point. Their graphs look like either one of these graphs:
Graph does not touch or intersect the x-axis and faces up. This function can't be written in intercept form since it does not have real roots.

Graph does not touch or intersect the x-axis and faces down. This function can't be written in intercept form since it does not have real roots.

These quadratic functions can not be written in the intercept form because we don't have any real number to put for in place of 'p' and 'q' in their intercept form.

Note on complex/imaginary roots

When a quadratic function does not have any real root, it can have imaginary roots. For example, the function `y = x^2 + 1` does not have any real roots, but it's imaginary roots are `i` and `-i`. Thus, if we put `p = i` and `q = -i` in the intercept form, we can write the function as follows:
`y = (x - i)(x + i)`
Thus, if we involve complex numbers (or imaginary numbers), all quadratic functions can be written in intercept form. On the other hand, if we consider only real numbers, a quadratic function can only be written in intercept form if it has one or two real roots.

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