Methods for Solving….. Factoring Factoring Square Roots Square Roots Completing the Square Completing the Square Graphing Graphing Quadratic Formula Quadratic Formula

Basic Principles Standard Form of a Quadratic Equation Standard Form of a Quadratic Equation ax 2 + bx + c = 0 Note: It is important to set the equation equal to zero for solving.  ax 2 is the quadratic term  bx is the linear term  c is the constant term

What does it mean to SOLVE a quadratic equation? Solution Solution Roots Roots Zeros Zeros “What x is….” “What x is….” x = x =

How many solutions and what kind of solutions? 0, 1, or 2 real 0, 1, or 2 real not real?!?!!!?? not real?!?!!!?? Imaginary solutions… Imaginary solutions… –remember i

Solutions: Visually (Graphically) Roots or Zeros Roots or Zeros

Factoring This method is generally the easiest as well as the most used in Algebra classrooms, mainly due to the fact that factoring is used in many future applications. This method is generally the easiest as well as the most used in Algebra classrooms, mainly due to the fact that factoring is used in many future applications. Also because there are two special cases worth memorizing. Also because there are two special cases worth memorizing.

Solve by Factoring Example 1 Example 1 x 2 + 6x – 7 = 0 (x + 7)(x – 1)=0 either… x + 7 = 0 or x – 1 = 0 so… x = - 7 or x = 1 This method involves several principles. 1.the F.O.I.L. method 2.the outer and inner terms must have a sum of the linear term 3.the signs must confirm the multiplication 4.Zero Product Property

Special Case 1 “A perfect squared trinomial” “The first term squared plus twice the first times the last plus the last term squared.” “The first term squared plus twice the first times the last plus the last term squared.” (first) 2 + 2(first)(last) + (last) 2 (first) 2 + 2(first)(last) + (last) 2 (x) 2 + 2(x)(7) + (7) 2 (x) 2 + 2(x)(7) + (7) 2 x x + 49 x x + 49 The last line is a great example of a “perfect squared trinomial.

Factoring a Perfect Squared Trinomial Since the middle term is the sum of the inner and outer terms (F.O.I.L.), and it is twice the first times the last….the inner and outer must be the same. Since the middle term is the sum of the inner and outer terms (F.O.I.L.), and it is twice the first times the last….the inner and outer must be the same. The only way this can happen is when the terms are identical… The only way this can happen is when the terms are identical… (a + b)(a + b) or a 2 + ab + ab +b 2 (a + b)(a + b) or a 2 + ab + ab +b 2

Factoring Perfect Squared Trinomials x x + 25 = 0 x x + 25 = 0 (x + 5) 2 = 0 (x + 5) 2 = 0 x + 5 = 0 x + 5 = 0 x = -5 x = -5

Special Case 2 “A Difference of Squares” a 2 – b 2 a 2 – b 2 (a + b)(a – b) (a + b)(a – b) Notice here that the inner and outer terms will be opposites and therefore cancel each other out when the F.O.I.L. method is used. Notice here that the inner and outer terms will be opposites and therefore cancel each other out when the F.O.I.L. method is used.

Using Difference of Squares to Solve Quadratic Equations x 2 – 25 = 0 x 2 – 25 = 0 (x + 5)(x – 5) = 0 (x + 5)(x – 5) = 0 Either… Either… X + 5 = 0 or x – 5 = 0 x = - 5 x = 5 x = - 5 x = 5

Solving by using Square Roots

Example: Square Root Method

Completing the Square A method that requires deeper understanding of perfect squared trinomials. A method that requires deeper understanding of perfect squared trinomials. Key question: What is the relationship of the linear coefficient and the constant? Key question: What is the relationship of the linear coefficient and the constant? Key answer: “Half of the middle term squared.” Key answer: “Half of the middle term squared.”

Completing the Square, continued Use a 2 + 2ab + b 2 Use a 2 + 2ab + b 2

Example Complete the Square x x + ____ x x + ____

Solving Using Completing the Square x x – 7 = 0 x x – 7 = 0

Solving Using Completing the Square x x – 7 = 0 x x – 7 = 0

Using Completing the Square to rewrite into Vertex Form y = ax 2 + b x + c y = ax 2 + b x + c y = a(x – h) 2 + k y = a(x – h) 2 + k

Vertex Form Example y = x 2 + 4x + 3 y = x 2 + 4x + 3

Vertex Form Example y = x 2 + 4x + 3 y = x 2 + 4x + 3 y = (x + 2) 2 – 1 y = (x + 2) 2 – 1

THE QUADRATIC FORMULA Basic Principle: It would not be called the FORMULA if it did not work ALWAYS! Basic Principle: It would not be called the FORMULA if it did not work ALWAYS! (does the formula for area of a triangle always work, does the pythagorean theorem always work)---YES!!!! (does the formula for area of a triangle always work, does the pythagorean theorem always work)---YES!!!!

THE QUADRATIC FORMULA if ax 2 + bx + c = 0, then…

The Discriminant b 2 – 4ac b 2 – 4ac this is the part under the radical in the quadratic formula this is the part under the radical in the quadratic formula it is telling because, taking the square root of a negative number, positive number, or zero create interesting scenarios. it is telling because, taking the square root of a negative number, positive number, or zero create interesting scenarios.

Rules and Roots if discriminant equals… if discriminant equals… ZERO ZERO –then there is only ONE REAL ROOT POSITIVE POSITIVE –then there are two roots, both are REAL RATIONAL or REAL IRRATIONAL NEGATIVE NEGATIVE –then there are two IMAGINARY ROOTS

MORE RULES CHOOSING A METHOD CHOOSING A METHOD –USE THE DISCRIMINANT TO HELP THE DECISION  Perfect square –factor  Zero –factor  Positive non-perfect square –quadratic formula