Algebra & Manipulation

For $ax^2+bx+c=0$ with roots $r,s$: sum $r+s = -\dfrac ba$ and product $rs = \dfrac ca$.

In plain terms. The two numbers that solve a quadratic add up to $-b/a$ and multiply to $c/a$ — you can read those off without solving.

Example. For $x^2 - 7x + 12 = 0$ the roots sum to $7$ and multiply to $12$, so they are $3$ and $4$.

A ratio $a:b$ means the amounts are $ak$ and $bk$. A proportion $\dfrac ab = \dfrac cd$ cross-multiplies to $ad = bc$.

In plain terms. A ratio is a shared multiplier — turn "$3$ to $5$" into "$3k$ and $5k$," then use the given total to find $k$.

Example. If red to blue is $3:5$ and there are $40$ total, then $3k + 5k = 40$, so $k = 5$: $15$ red and $25$ blue.

The expression $xy + ax + by$ factors as $(x+b)(y+a) - ab$.

In plain terms. When an equation mixes a product like $xy$ with plain $x$ and $y$ terms, add the one missing number to make it factor into a neat product.

Example. For $xy + 2x + 3y = 8$, add $6$: $(x+3)(y+2) = 14$, so the integer factor pairs of $14$ give the solutions.