Lesson 7.3: Quadratic Equations and Functions

Introduction

In this lesson, students will delve into the fascinating world of quadratic equations and functions. Quadratic equations play a crucial role in various areas of mathematics and are essential for the GRE General Test. Our objectives will be to learn how to solve quadratic equations using factoring and the quadratic formula, understand functions and their notation, and recognize common factoring patterns.

Learning Objectives

• Solve quadratic equations by factoring and the quadratic formula.
• Understand functions, domain, and function notation.
• Recognize common factoring patterns.
• Solve quadratic equations by appropriate methods.
• Evaluate and interpret functions and their notation.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two that can be written in the standard form:

$$

ax^2 + bx + c = 0

$$

where:

• $a$, $b$, and $c$ are constants, with $a \neq 0$.
• $x$ represents the variable or unknown.

Characteristics of Quadratic Equations

• The graph of a quadratic equation is a parabola.
• If $a > 0$, the parabola opens upward, and if $a < 0$, it opens downward.
• The vertex of the parabola represents the maximum or minimum point, depending on the direction it opens.

Solving Quadratic Equations by Factoring

One method of solving quadratic equations is by factoring. This involves expressing the quadratic equation in factored form:

$$

(a x + r)(b x + s) = 0

$$

To apply this method, follow these steps:

1. Move all terms to one side of the equation: $ax^2 + bx + c = 0$.
2. Factor the quadratic expression on the left side.
3. Set each factor equal to zero and solve for $x$.

Example 1: Factoring a Quadratic Equation

Consider the quadratic equation:

$$

x^2 - 5x + 6 = 0

$$

1. We look for two numbers that multiply to $6$ (the constant term) and add to $-5$ (the coefficient of $x$). The numbers are $-2$ and $-3$.
2. We can factor the equation as:

$$

(x - 2)(x - 3) = 0

$$

1. Now, set each factor to zero:

• $x - 2 = 0 \Rightarrow x = 2$
• $x - 3 = 0 \Rightarrow x = 3$

1. Thus, the solutions to the equation are $x = 2$ and $x = 3$.

Common Factoring Patterns

Understanding common factoring patterns can greatly simplify the process:

1. Difference of squares:

• $$a^2 - b^2 = (a - b)(a + b)$$

1. Perfect square trinomial:

• $$a^2 + 2ab + b^2 = (a + b)^2$$
• $$a^2 - 2ab + b^2 = (a - b)^2$$

1. Sum and difference of cubes:

• $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$
• $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

Solving Quadratic Equations Using the Quadratic Formula

The quadratic formula is a universal method that applies to all quadratic equations:

$$

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

$$

The term under the square root, $b^2 - 4ac$, is called the discriminant. It determines the nature of the roots:

• If $b^2 - 4ac > 0$, there are two distinct real roots.
• If $b^2 - 4ac = 0$, there is one real root (a repeated root).
• If $b^2 - 4ac < 0$, there are no real roots (the roots are complex).

Example 2: Using the Quadratic Formula

Let's solve the equation:

$$

x^2 - 4x + 4 = 0

$$

1. Identify $a = 1$, $b = -4$, and $c = 4$.
2. Compute the discriminant:

$$

D = (-4)^2 - 4(1)(4) = 16 - 16 = 0

$$

1. Since $D = 0$, we have one repeated root. Now, apply the quadratic formula:

$$

x = $\frac{-(-4) \pm \sqrt{0}}{2(1)}$ = $\frac{4}{2}$ = 2

$$

Thus, the solution is $x = 2$, which is a repeated root.

Understanding Functions

A function is a relationship between two sets that assigns exactly one output for each input. Functions can be expressed as equations, tables, or graphs.

Function Notation

Function notation is typically written as:

$$

$f(x) = mx + b$

$$

where:

• $f(x)$ denotes the function dependent on $x$.
• $m$ represents the slope, and $b$ represents the y-intercept.

Example 3: Evaluating a Function

Consider the function:

$$

$f(x) = 2x + 3$

$$

To evaluate this function at $x = 4$:

$$

f(4) = 2(4) + 3 = 8 + 3 = 11

$$

Thus, $f(4) = 11$ indicates the output when the input is $4$.

Conclusion

In this lesson, students explored quadratic equations, learned to solve them by factoring and using the quadratic formula, and understood basic function concepts. Quadratic equations are an essential part of algebra that will appear in various practical scenarios and tests like the GRE.

Study Notes

• Quadratic equations have the form $ax^2 + bx + c = 0$.
• To solve by factoring, express the equation as a product of factors and set them to zero.
• The quadratic formula is $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Use it for any quadratic equation.
• The discriminant $D = b^2 - 4ac$ helps determine the nature of the roots of the equation.
• Function notation $f(x)$ is used to define functions, where $x$ is the input and $f(x)$ is the output.