Factoring and Quadratics
What you must understand
Factoring is the reverse of multiplication, and the standard sequence is: greatest common factor first, then patterns (difference of squares, perfect-square trinomials), then trinomial factoring into two binomials, then grouping for four-term polynomials. The zero-product property says a product is zero only if a factor is zero, which turns a factored quadratic into simple linear equations. Quadratics can be solved by factoring, completing the square, or the quadratic formula x = [-b +/- sqrt(b^2 - 4ac)]/(2a). The discriminant b^2 - 4ac tells you the number of real solutions: positive gives two, zero gives one, negative gives none. The vertex of y = ax^2 + bx + c sits at x = -b/(2a).
Traps to watch
- Forgetting the GCF, leaving answers unfactored.
- Sign errors when the constant term is negative: the two binomials need opposite signs.
- Dropping a solution by dividing both sides by x instead of factoring it out.
- Arithmetic slips inside the discriminant, especially squaring a negative b.
- Assuming a negative discriminant means "no answer" rather than "no real solutions."
Worked example
Solve x^2 - 5x - 14 = 0. Find two numbers multiplying to -14 and adding to -5: -7 and 2. Factor as (x - 7)(x + 2) = 0, so x = 7 or x = -2.
Rule of thumb
Always pull the GCF first, and always set the equation equal to zero before applying the zero-product property.