Factoring and Solving Quadratic Equations: Intermediate Algebra Study Notes

Factoring Polynomials

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials with four or more terms. The terms are grouped in pairs, and common factors are factored out from each group.

• Key Point 1: Group terms with common factors.

• Key Point 2: Factor out the greatest common factor from each group.

• Key Point 3: If the resulting binomials are the same, factor them out.

• Example: $4m^2 + 20mn + 25n^2 = (2m)^2 + 2 \times 2m \times 5n + (5n)^2 = (2m + 5n)^2$

Factoring the Difference of Squares

The difference of squares is a special factoring pattern: $a^2 - b^2 = (a + b)(a - b)$.

• Key Point 1: Recognize the form $a^2 - b^2$.

• Key Point 2: Factor as $(a + b)(a - b)$.

• Example: $y^2 - 81 = (y + 9)(y - 9)$

Factoring Trinomials

Trinomials of the form $ax^2 + bx + c$ can often be factored into two binomials.

• Key Point 1: Find two numbers that multiply to $ac$ and add to $b$.

• Key Point 2: Rewrite the middle term and factor by grouping.

• Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$

Solving Quadratic Equations by Factoring

Zero Product Property

The zero product property states that if $ab = 0$, then $a = 0$ or $b = 0$. This is used to solve equations after factoring.

• Key Point 1: Set the equation equal to zero.

• Key Point 2: Factor the quadratic expression.

• Key Point 3: Set each factor equal to zero and solve for the variable.

• Example: $(x - 3)(x + 5) = 0 \Rightarrow x - 3 = 0$ or $x + 5 = 0 \Rightarrow x = 3$ or $x = -5$

Quadratic Equations in Standard Form

A quadratic equation in one variable has the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

• Key Point 1: Rearrange the equation to standard form.

• Key Point 2: Factor the quadratic expression.

• Key Point 3: Apply the zero product property to solve.

• Example: $x^2 - 81 = 0 \Rightarrow (x + 9)(x - 9) = 0 \Rightarrow x = 9$ or $x = -9$

Solving Quadratic Equations by the Square Root Property

Square Root Property

If $x^2 = a$, then $x = \pm \sqrt{a}$. This property is used to solve quadratic equations that can be written in the form $x^2 = a$.

• Key Point 1: Isolate the squared term.

• Key Point 2: Take the square root of both sides, remembering to include both positive and negative roots.

• Example: $x^2 = 100 \Rightarrow x = \pm \sqrt{100} \Rightarrow x = \pm 10$

Quadratic Functions and the Fundamental Theorem of Algebra

Quadratic Functions

A quadratic function is a function of the form $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of a quadratic function is a parabola.

• Key Point 1: The roots of the quadratic function are the values of $x$ for which $f(x) = 0$.

• Key Point 2: The roots can be found by factoring, completing the square, or using the quadratic formula.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-zero, single-variable polynomial of degree $n$ has exactly $n$ roots, counting multiplicity.

• Key Point 1: For quadratic equations ($n = 2$), there are two roots.

• Key Point 2: Roots may be real or complex.

Summary Table: Methods for Solving Quadratic Equations

The following table summarizes the main methods for solving quadratic equations:

Method	Form of Equation	Example
Factoring	$ax^2 + bx + c = 0$	$x^2 + 5x + 6 = 0$
Square Root Property	$x^2 = a$	$x^2 = 100$
Quadratic Formula	$ax^2 + bx + c = 0$	$x^2 + 2x + 1 = 0$
Completing the Square	$ax^2 + bx + c = 0$	$x^2 + 6x + 9 = 0$

Additional info: These notes cover factoring, solving quadratic equations by factoring and the square root property, and introduce quadratic functions and the Fundamental Theorem of Algebra, all of which are core topics in Intermediate Algebra.