Factoring & Quadratic Equations Cheat Sheet

Mastering factoring and quadratic equations is key to solving polynomial expressions and understanding their roots. This cheat sheet covers essential techniques and formulas.

Core Principles

Factoring: Breaking down an expression into a product of simpler expressions.
Quadratic Equation: An equation of the form ax^2 + bx + c = 0, where a ≠ 0.
Roots/Solutions: The values of x that satisfy the quadratic equation.
The relationship between factoring and finding roots is direct: factors reveal the roots.
Completing the square is a method to transform any quadratic into a solvable form.
The quadratic formula provides a universal solution for any quadratic equation.

Action Steps

1. Identify the type of expression or equation.
2. For factoring, look for common factors first.
3. Apply appropriate factoring techniques (e.g., grouping, difference of squares).
4. For quadratic equations, set the equation to zero.
5. Choose a solution method: factoring, completing the square, or quadratic formula.
6. If factoring, find factors that yield roots.
7. If using the quadratic formula, substitute coefficients carefully.
8. Simplify the results and check your solutions.

Formulas

Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant: $\Delta = b^2 - 4ac$
Factoring Trinomials (ax^2 + bx + c): Find two numbers that multiply to 'ac' and add to 'b'.
Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$
Perfect Square Trinomial: $a^2 + 2ab + b^2 = (a + b)^2$
Perfect Square Trinomial: $a^2 - 2ab + b^2 = (a - b)^2$

Key Terms

Trinomial: A polynomial with three terms.
Binomial: A polynomial with two terms.
Root: A value of the variable that makes an equation true.
Coefficient: The numerical factor of a term.
Constant: A term without a variable.
Discriminant: The part of the quadratic formula under the square root ($b^2 - 4ac$), indicating the nature of the roots.

Quiz

What is the first step in factoring any polynomial?: Look for a common factor
If the discriminant ($b^2 - 4ac$) is positive, how many real roots does the quadratic equation have?: Two