Quadratic equations: Parabola
Quadratics
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} #
#a=# #-1#
#b=# #-14#
#c=# #-41#
When we compare #y=-x^2-14\cdot x-41# with #y=ax^2+bx+c#, we find
#a=# #-1#
#b=# #-14#
#c=# #-41#
#b=# #-14#
#c=# #-41#
When we compare #y=-x^2-14\cdot x-41# with #y=ax^2+bx+c#, we find
#a=# #-1#
#b=# #-14#
#c=# #-41#
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