Book cover for Elementary Algebra

Elementary Algebra

Lynn Marecek

ISBN #9780998625713

1st Edition

6,645 Questions

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32,192 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter covers methods for solving quadratic equations beyond simple factoring. Key techniques include using the Square Root Property for equations of the form x² = k and a(x – h)² = k, as well as the method of completing the square to turn a general quadratic into a perfect square trinomial. These methods not only provide two solutions (when applicable) but also demonstrate how quadratic equations can model real-world situations like the trajectory of fireworks.

Learning Objectives

1

Solve quadratic equations of the form ax² = k using the Square Root Property.

2

Solve quadratic equations of the form a(x – h)² = k by isolating the squared term and applying the Square Root Property.

3

Complete the square of a binomial expression to transform a quadratic equation into a perfect square form.

4

Analyze and model real-world situations, such as the trajectory of fireworks, with quadratic equations.

Key Concepts

CONCEPT

DEFINITION

Quadratic Equation

An equation of the form ax² + bx + c = 0 (with a ≠ 0) which includes a variable squared, requiring specialized solving methods.

Square Root Property

A property stating that if x² = k (and k ≥ 0), then x = ±√k. It is used to solve equations where the variable is squared.

Completing the Square

A method to convert a quadratic expression into a perfect square trinomial of the form (x – h)², enabling easier solution using the Square Root Property.

Perfect Square Trinomial

A quadratic expression that factors into (x ± d)², where the expression matches the pattern a² + 2ab + b².

Example Problems

Example 1

In the following exercises, solve the following quadratic equations. $$ a^{2}=49 $$

Example 2

In the following exercises, solve the following quadratic equations. $b^{2}=144$

Example 3

In the following exercises, solve the following quadratic equations. $r^{2}-24=0$

Example 4

In the following exercises, solve the following quadratic equations. $t^{2}-75=0$

Example 5

In the following exercises, solve the following quadratic equations. $u^{2}-300=0$
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Step-by-Step Explanations

QUESTION

How can you solve the equation x² = 169 using the Square Root Property?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the equation is of the form x² = k, where k = 169.
Step 2: Apply the Square Root Property: x = ±√169.
Step 3: Simplify the square root: √169 = 13.
Step 4: Write the final solutions: x = 13 and x = -13.
Final Answer: x = 13, x = -13

Solving x² = 169 using the Square Root Property

QUESTION

How do you solve the quadratic equation (x – 3)² = 16?

STEP-BY-STEP ANSWER:

Step 1: Identify that the equation is already in the form (x – h)² = k with h = 3 and k = 16.
Step 2: Apply the Square Root Property: x – 3 = ±√16.
Step 3: Simplify the square root: √16 = 4.
Step 4: Set up two separate equations: x – 3 = 4 and x – 3 = -4.
Step 5: Solve each equation: For x – 3 = 4, x = 7; for x – 3 = -4, x = -1.
Final Answer: x = 7, x = -1

Solving (x – 3)² = 16 using the Square Root Property

QUESTION

How do you solve x² + bx + c = 0 by completing the square?

STEP-BY-STEP ANSWER:

Step 1: Rearrange the equation so that the constant term is on the opposite side: x² + bx = -c.
Step 2: Find the value needed to complete the square: (b/2)².
Step 3: Add (b/2)² to both sides of the equation.
Step 4: Write the left side as a perfect square trinomial: (x + b/2)².
Step 5: Apply the Square Root Property to solve for x.
Final Answer: x = (-b/2) ± √((b/2)² - c)

Completing the Square

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Common Mistakes

  • Failing to isolate the quadratic term before applying the Square Root Property.
  • Neglecting to adjust the coefficient to 1 by dividing when the quadratic term has a coefficient other than 1.
  • Forgetting to consider both the positive and negative roots when taking the square root of both sides.
  • Incorrectly completing the square by using the wrong value for (b/2)².
  • Overlooking domain restrictions, especially in cases where the constant k is negative leading to no real solutions.