Quadratics

Quadratic equations: Parabola

Quadratics

A formula is called a quadratic if it has the form \[y=\blue ax^2+\green bx+\purple c\] where #\blue a#, #\green b# and #\purple c# are numbers and #\blue a\ne0#.

Example

\[\begin{array}{rcl}y&=&\blue 3x^2\green{-2}x+\purple 3
\end{array}\]

Reducing to standard form

Quadratics can be expressed in a number of different ways. By expanding the brackets, it becomes clear whether the formula is indeed a quadratic, and what the values of #\blue a#, #\green b# and #\purple c# are.

Example

# \begin{array}{rcl}y&=&\left(2x+2\right)\left(x+3\right)\\ &=& 2x^2+x \cdot 2 +2x\cdot 3+2 \cdot 3 \\ &=& \blue 2x^2+\green 8x+\purple 6 \end {array} #

Take a look at the formula #y=8\cdot x^2-16\cdot x+15#.

Give the values of #a#, #b# and #c# when we write the fomula in the form #y=ax^2+bx+c#.

#a=# #8#
#b=# #-16#
#c=# #15#

When we compare #y=8\cdot x^2-16\cdot x+15# with #y=ax^2+bx+c#, we find
#a=# #8#
#b=# #-16#
#c=# #15#

New example