Revision sheet

Solving by square roots

Isolate the square; then $\pm\sqrt{\;}$.

Watch out: two solutions, not one.

Zero product property

Product = 0: set each factor to 0.

Watch out: never divide by $x$; you lose $x = 0$.

Solving by factoring

Multiply to $c$, add to $b$; then each factor $= 0$.

Watch out: the factor $x - 4$ gives $x = 4$.

Completing the square

Add $(b/2)^2$ to both sides; then square roots.

Watch out: halve, then square; and do it on both sides.

The quadratic formula

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$; the discriminant counts the solutions.

Watch out: $2a$ divides the whole numerator.

Modeling with quadratic equations

Translate, solve, then keep the solutions that fit the situation.

Watch out: a negative length or the start time is not the answer.

Solving for a variable

Same moves as with numbers; the other letters are just numbers you do not know.

Watch out: divide every term.

A line and a conic

1. Substitute the line into the conic.
2. Solve the quadratic: 0, 1 or 2 real solutions.
3. Find the other coordinate from the line.

Watch out: a negative square means no intersection.

Complex solutions

Discriminant < 0 → $x = p \pm qi$ (conjugates).

Watch out: divide the whole numerator by $2a$.