### 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=# #-5#

#b=# #-10#

#c=# #-9#

When we compare #y=-5\cdot x^2-10\cdot x-9# with #y=ax^2+bx+c#, we find

#a=# #-5#

#b=# #-10#

#c=# #-9#

#b=# #-10#

#c=# #-9#

When we compare #y=-5\cdot x^2-10\cdot x-9# with #y=ax^2+bx+c#, we find

#a=# #-5#

#b=# #-10#

#c=# #-9#

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