Math 2 Honors: Quadratics & Factoring Review

First step before factoring a quadratic equation

Move all terms to one side of the equation to set it equal to $0$

Standard form of a quadratic equation

$$ax^2 + bx + c = 0$$

Sign rule when moving a term across the equals sign

Change its sign

Rule for combining like terms in quadratic equations

Combine $x$ terms with $x$ terms and constant terms with constant terms

Simple factoring goal for $x^2 + bx + c = 0$

Find two numbers that multiply to $c$ and add up to $b$

Sign pattern when the last term ($c$) is positive

Both signs in the factors are the same

Sign pattern when the last term ($c$) is negative

The signs in the factors are opposite

Factor sign rule when the middle term ($b$) is positive and signs are opposite

The factor with the larger absolute value is positive

Factor sign rule when the middle term ($b$) is negative and signs are opposite

The factor with the larger absolute value is negative

Factored form of a quadratic equation

$$(x + m)(x + n) = 0$$

Zero Product Property

Set each factor equal to $0$ and solve for the variable

Solution to $x + a = 0$

$$x = -a$$

Solution to $x - a = 0$

$$x = a$$

Initial step when the leading coefficient $a \neq 1$

Multiply $a \times c$, then find two numbers that multiply to $ac$ and add up to $b$

AC method

Factoring method where you multiply $a \times c$, split the middle term, group terms, and factor

Splitting the middle term

Rewrite $bx$ as two separate terms using the pair of numbers found from the AC method

Grouping method

Group the first two terms together and the last two terms together, then factor out the common factor from each group

GCF

Greatest Common Factor; the largest factor shared by all terms in an expression

Factor by grouping process

Factor the GCF out of each group, then factor out the shared binomial

Checking factored solutions

FOIL (multiply) the binomial factors to ensure they expand back to the original equation

Next step when no integer factor pair exists

Apply the quadratic formula

Steps for solving a quadratic equation by factoring

Move terms to one side $\rightarrow$ Combine like terms $\rightarrow$ Factor $\rightarrow$ Set each factor to $0$ $\rightarrow$ Solve $\rightarrow$ Check

Quadratic formula

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Role of $a$ in standard quadratic form

The coefficient of $x^2$

Role of $b$ in standard quadratic form

The coefficient of $x$

Role of $c$ in standard quadratic form

The constant term

Discriminant

$$b^2 - 4ac$$

Number of real solutions when the discriminant is positive

Two real solutions

Number of real solutions when the discriminant is zero

One real solution

Number of real solutions when the discriminant is negative

No real solutions