Lesson 76: Finding Roots of Quadratic Equations

Introduction to the Final Quarter of ShoreMont Mathematics Algebra One

• Context and Course Progression:

  • Lesson 76 marks the beginning of the fourth and final quarter of ShoreMont Mathematics Algebra one.

  • There are 25 lessons remaining in the course (Lessons 76 through 100).

  • The instructor encourages students to "finish strong" and complete these last 25 lessons to the best of their ability.

• The Role of Attitude in Learning:

  • The instructor emphasizes the importance of a "God glorifying attitude" while performing schoolwork.

  • Work should reveal an attitude of thankfulness and gratefulness for the opportunity to learn.

  • Learning mathematics is presented as a means to learn more about God and His creation.

  • Mathematics as a Language: Mathematics is defined as the "language of science" and a tool for studying creation.

  • A poor attitude (complaining or whining) is viewed as a barrier to the joy of education, regardless of the quality of the math program.

Concepts and Definitions of Quadratic Equations

• Building on Previous Material:

  • Lesson 76 builds directly upon Lesson 75, which covered identifying and factoring quadratic equations.

• Standard Form of Quadratic Equations:

  • The standard form is expressed as: $ax^2 + bx + c = 0$.

• Understanding the Set Value of Zero:

  • Common questions arise regarding why quadratic equations are set to zero rather than another number like $1$.

  • Quadratic equations can also be expressed as functions or y-values:

    • $y = ax^2 + bx + c$

    • $f(x) = ax^2 + bx + c$

  • The standard form is simply the specific case where $y$ or $f(x)$ is equal to zero.

• Applications and Importance of Zeros:

  • Graphical Representation: Quadratic equations represent parabolas when graphed on an $x y$ axis.

  • Real-World Applications: Quadratic equations are used to model motion, specifically acceleration.

  • Significance of Zeros: The values of $x$ when $y$ or $f(x)$ equals zero (the roots) are especially important in physical applications.

  • Ease of Calculation: It is mathematically easier to learn how to solve quadratic relationships when they are set to zero rather than complex numbers like $10.2$.

• Key Terminology:

  • Root of a Polynomial: This is synonymous with the term "zero."

  • The terms "root" and "zero" are used interchangeably in this lesson.

The Zero Factor Theorem

• Definition:

  • If $a$ and $b$ are real numbers such that $a \times b = 0$, then either $a = 0$, $b = 0$, or both are zero.

• Metaphorical Explanation ("Bag of Rocks"):

  • When looking at a factored equation like $(x + 3)(x + 2) = 0$, think of the terms inside each set of parentheses as a separate "bag of rocks."

  • Consider $(x + 3)$ as "Bag A" and $(x + 2)$ as "Bag B."

  • To make the product of the bags zero, at least one of the bags must contain zero.

Procedures for Finding Roots

• The Factoring Method:

  • Step 1: Ensure the equation is in standard form ($ax^2 + bx + c = 0$).

  • Step 2: Factor the trinomial into two binomials.

  • Step 3: Set each binomial equal to zero and solve for $x$.

• The "Opposite Constant" Shortcut:

  • Instead of setting up formal equations, one can look at the constant term within the binomial and take its opposite.

  • Example: For the binomial $(x + 3)$, the root is $-3$, because $-3 + 3 = 0$.

  • Example: For the binomial $(x - 1)$, the root is $1$, because $1 - 1 = 0$.

Example 76.1: Finding Roots/Zeros

• Problem A:

  • Equation: Specified as the factored form of the first quadratic in lesson 75.2 ($x^2 + 5x + 6 = 0$).

  • Factoring: $(x + 3)(x + 2) = 0$ (or alternatively $(x + 2)(x + 3) = 0$, as order does not matter).

  • Application of Zero Factor Theorem:

    • $x + 3 = 0 \rightarrow x = -3$

    • $x + 2 = 0 \rightarrow x = -2$

  • Solutions: $x = -3, -2$.

• Problem B:

  • Equation: $x^2 + 2x - 3 = 0$

  • Factoring: $(x + 3)(x - 1) = 0$

  • Mental Calculation: The opposite of $+3$ is $-3$, and the opposite of $-1$ is $+1$.

  • Solutions: $x = -3, 1$.

• Problem C (Equation Preparation):

  • Initial form: $x^2 + 10x = -25$

  • Standard Form Conversion: Add $25$ to both sides to get $x^2 + 10x + 25 = 0$.

  • Factoring: $(x + 5)(x + 5) = 0$ or $(x + 5)^2 = 0$.

  • Solution: $x = -5$. (There is only one solution in this case because the binomials are identical).

• Problem D (Cubic with Quadratic Relationship):

  • Equation: $x^3 - 6x^2 + 8x = 0$

  • Initial Step: Factor out the greatest common factor, which is $x$.

  • Intermediate Form: $x(x^2 - 6x + 8) = 0$

  • Factoring the Quadratic: $x(x - 4)(x - 2) = 0$

  • Identifying Roots:

    • From the lone factor $x$: $x = 0$

    • From $(x - 4)$: $x = 4$

    • From $(x - 2)$: $x = 2$

  • Solutions: This equation has three roots: $0, 4, 2$.

  • Note: This demonstrates that the Zero Factor Theorem applies even to equations that are not strictly quadratic, provided they can be factored.