Can a quadratic be written in all three forms (standard, vertex and intercept)?

There are three forms in which a quadratic equation/expression can be written.

  • Standard form (`ax^2 + bx + c = 0`)
  • Vertex form (`a(x - h)^2 + k = 0`)
  • Intercept form (`a(x - p)(x - q) = 0`)

All quadratic equations or expressions can not be written in all the three forms, though they all can be written in the standard form and vertex form.

This is because the intercept form of a quadratic expression is its factored form, and not all quadratic expressions are factor-able  This however does not have any relation with the parabola of a quadratic having or not having x-intercepts. It depends solely on whether a quadratic expression can be factored or not.

If a quadratic expression can not be factored into a product of lesser degree binomials, then it can not be written in the intercept form.

Quadratic equations/functions whose parabolas do not touch or intersect the x-axis do not have any real roots and so they can't be factored into the intercept form. This, however, does not imply that a quadratic equation whose parabola has x-intercepts can always be written in the intercept form, because there are quadratic equations which do have x-intercepts but can't be factored by the method of splitting the middle term.

For example `x^2 + 3x + 1` is a quadratic expression which can not be factored, so it can't be written in the intercept form. It can be written in the vertex form `(x + 3/2)^2 - 5/4`. It's graph does have x-intercepts, so don't confuse the concept that if a parabola has x-intercepts it's equation can be written in intercept form. However, if a parabola does not have x-intercepts then it can't be written in the intercept form.

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