Forms of Quadratics

You can represent a quadratic equation in a whole lot of different forms. Each has its advantages.

Standard forms let you learn valuable information about an equation just by looking at what number was put where. There are three common forms of quadratic equations, also known as second-order polynomials.

Note that variables like a, b, and c in these descriptions will change in meaning from form to form.

Standard form

$$y=\text{ax}^2+\text{bx}+c$$

Advantage: Relative mathematical simplicity, sets you up to use the quadratic formula, and easy to find the y-intercept. It is just the constant y=c.

Example: $$y=2 x^2+12 x-14$$

Factored form

$$y=a(x+b)(x+c)$$

Advantage: Easy to find the x-intercepts. They are at x=-b and x=-c. You don’t even need the quadratic formula!

Example $$y = 2(x+7)(x-1)$$

Vertex form

$$y=a(x-h)^2+k$$

Advantage: Easy to find the vertex, which is at (h,k).

Example: $$y=2 (x+3)^2-32$$

All three of the examples from this post were actually the same quadratic equation! See if you can identify the vertex, x-intercepts, and y-intercepts using the different forms.

Author: Wade

Materials scientist, lab manager, and educator based in the Seattle area.

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