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: $$(x+p)(x+q)=x\cdot x+x\cdot q+p\cdot x+p\cdot q$$
3. Combine like terms: $$(x+p)(x+q)=x^2+(p+q)x+pq$$

Standard Form And Factor Form Are Connected By Patterns

1. Quadratics are often written in standard form: $$ax^2+bx+c$$
2. and sometimes in factor form: $$(x+p)(x+q)$$
3. When the factor form is exactly $(x+p)(x+q)$, the leading coefficient is $a=1$.
4. The patterns are:
   1. $a=1$
   2. $b=p+q$
   3. $c=pq$

Factorizing Means Rewriting A Quadratic As A Product

1. Factorizing is the reverse of expanding.
2. You are trying to "undo" the multiplication.

Factorizing When $a = 1$

1. To factorize $x^2+bx+c$, you look for two numbers $p$ and $q$ such that:
   1. $p+q=b$
   2. $pq=c$
2. Then $$x^2+bx+c=(x+p)(x+q)$$

Special Pattern: Difference Of Two Squares

A very important quadratic pattern is the difference of two squares: $$a^2-b^2=(a-b)(a+b)$$

Factorizing When $a\neq 1$

1. When the coefficient of $x^2$ is not 1, you often factorize by finding two numbers whose product is $ac$ and whose sum is $b$.
2. For $ax^2+bx+c$, steps are:
   1. Compute $ac$.
   2. Find two numbers $m$ and $n$ such that $mn=ac$ and $m+n=b$.
   3. Rewrite $bx$ as $mx+nx$.
   4. Factorize by grouping.

Common Factor First: A Key Preparation Skill

1. Before factorizing a quadratic, always check for a common factor and factor that out first.
2. It makes the remaining quadratic simpler.

Checking Your Factorization By Expanding

After factorizing, you should quickly expand to verify.