MYP Standard Mathematics Quadratic Expressions Notes

1. A quadratic expression is one of the most important algebraic patterns you will meet in Standard Mathematics because it appears in models of area, motion, profit, and many other contexts.
2. To work confidently with quadratics, you need to be fluent in two opposite processes:
   1. Expanding (multiplying out brackets to remove them)
   2. Factorizing (rewriting an expression as a product of brackets)

Quadratic Expressions Are Defined By Their Highest Power

1. A useful way to recognize a quadratic is to look for the highest power of the variable.
   1. $x^2-5x-14$ is quadratic (highest power is 2).
   2. $3x^2+2x$ is quadratic.
   3. $2y-5y^2+7$ is quadratic (it still has a $y^2$ term, even if the order is mixed).
2. Not quadratic:
   1. $x^3-5x-14$ is cubic (highest power is 3).
   2. $8t+12$ is linear (highest power is 1).
   3. $x^2+3y-25$ is not a quadratic in one variable because it has two different variables.

Expanding Brackets Creates An Equivalent Expression

1. When you expand, you do not change the value of the expression, you only change its form.
2. In algebra we often show this with the symbol $\equiv$, meaning identically equal.

Expanding Two Binomials (FOIL / Grid Method)

1. A very common quadratic form comes from multiplying two binomials (two-term brackets): $$(x+p)(x+q)$$
2. Expand by multiplying every term in the first bracket by every term in the second: $$(x+p)(x+q)=x\cdot x+x\cdot q+p\cdot x+p\cdot q$$