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Factoring Trinomials

Overview:

If a trinomial $x^2+bx+c$ can be factored as $(x+p)(x+q)$ , then by expanding/foiling this expression we know that we need $ax^2+bx+c=x^2+(p+q)x+pq$ . Hence we know that in order to factor the trinomial we need two number $p,q$ such that the sum of the two is $b$ and the product is $c$ .

For the more complex problem, where the coefficient of $x^2$ is not 1, we can still use a similar method to the above.

Parallel and Perpendicular Lines

Overview:

Suppose we have to lines with the slopes $m_1$ and $m_2$ respectively. We can tell the relative position of two as following:

The two lines are parallel if $m_1=m_2$ (except when the two lines are vertical, because slope is undefined for vertical lines)
The two lines are perpendicular if $m_1\cdot m_2=-1$ (the proof can be seen here)

By using this informations one can solve problems involving finding the equation of line passing through a point and parallel (or perpendicular) to another line.

Solving Linear Equations