Find the Solutions to Your 8.3 Independent Practice Problems
This page provides the complete answer key for the “8.3 Independent Practice” from page 221, helping students check their work and understand the correct methods. Whether you are working on algebra, physics, or another subject, this guide will walk you through the solutions step-by-step.
Problem 1: Factoring a Quadratic Expression
Problem:
Factor the quadratic expression:
x2+7x+10x2+7x+10
Step-by-Step Solution:
Step 1: Understand the standard form.
A quadratic expression is usually written as ax2+bx+cax2+bx+c.
Here,
- a=1a=1 (the coefficient of x2x2)
- b=7b=7 (the coefficient of xx)
- c=10c=10 (the constant term)
Step 2: What are we looking for?
We want to write the quadratic as a product of two binomials:x2+7x+10=(x+m)(x+n)x2+7x+10=(x+m)(x+n)
where mm and nn are two numbers that:
- add to b=7b=7
- multiply to c=10c=10
Step 3: Find the factors of 10 that add up to 7.
Let’s list factor pairs of 10:
- 1 and 10 (1×10=101×10=10, 1+10=111+10=11)
- 2 and 5 (2×5=102×5=10, 2+5=72+5=7)
Step 4: Identify the correct pair.
2+5=72+5=7 (that’s our bb: check!)
2×5=102×5=10 (that’s our cc: check!)
Step 5: Write the answer.
So, the factorization is:x2+7x+10=(x+2)(x+5)x2+7x+10=(x+2)(x+5)
Final Answer:
(x+2)(x+5)(x+2)(x+5)
Problem 2: Solving a Quadratic Equation by Factoring
Problem:
Solve for xx:
x2−9=0x2−9=0
Step-by-Step Solution:
Step 1: Write the quadratic in standard form.
x2−9=0x2−9=0
This is already in the correct form: x2−9=0x2−9=0.
Step 2: Factor the expression.
Notice x2−9x2−9 is a difference of squares:x2−9=(x+3)(x−3)x2−9=(x+3)(x−3)
Step 3: Set each factor to zero (Zero Product Property).(x+3)=0or(x−3)=0(x+3)=0or(x−3)=0
Step 4: Solve each equation.x+3=0 ⟹ x=−3x+3=0⟹x=−3x−3=0 ⟹ x=3x−3=0⟹x=3
Final Answer:
x=−3orx=3x=−3orx=3
Problem 3: Percent Problem
Problem:
What is 45% of 320?
Step-by-Step Solution:
Step 1: Translate the problem into a mathematical equation.
To find 45% of 320 means to multiply 320 by 0.45 (since 45% as a decimal is 0.45).
Step 2: Perform the multiplication.45% of 320=0.45×32045% of 320=0.45×320
Step 3: Calculate.0.45×320=(0.45×300)+(0.45×20)=135+9=1440.45×320=(0.45×300)+(0.45×20)=135+9=144
Alternate method:
You can also multiply directly:0.45×320=1440.45×320=144
Final Answer:
144144
Deep-Dive on Understanding
Let’s explore why and how these problems are solved, what concepts are involved, and tips for similar problems.
Factoring Quadratics—Why Does It Matter?
Factoring transforms a quadratic from expanded to factored form. This is key when you want to:
- Solve quadratic equations
- Simplify algebraic expressions
- Understand the properties of parabolas (vertex, zeros/solutions, etc.)
Tip:
When you have a quadratic x2+bx+cx2+bx+c, always look for two numbers that:
- Multiply to cc
- Add to bb
Factoring is the reverse process of multiplying binomials (the FOIL method). Mastery here makes solving quadratics much easier!
Difference of Squares
The expression x2−9x2−9 is factorable using the formula:a2−b2=(a+b)(a−b)a2−b2=(a+b)(a−b)
So,
- a=xa=x
- b=3b=3
This is a common shortcut and underpins many solving strategies for quadratics where the middle term (bxbx) is missing.
Solving Percent Problems
Percent problems show up in all aspects of life—from sales, discounts, test scores, to data analysis.
To convert a percent to a decimal:
- Drop the percent sign
- Divide by 100
For 45%:45%=0.4545%=0.45
Finding “X% of Y” is always:(Decimal form of X%)×Y(Decimal form of X%)×Y
Common mistake:
Don’t multiply by the percent directly without converting! (E.g., not 45×32045×320, but 0.45×3200.45×320.)
Extending Your Understanding: Real-World Examples
Factoring in Context
Suppose you’re designing a rectangular garden, and you know the area and perimeter. Factoring help you determine possible dimensions that match your criteria!
Quadratic Equations in Action
Projectile motion in physics uses quadratic equations:
The height of a ball thrown upwards follows a quadratic formula. Finding when the height is zero (when the ball lands) is a quadratic equation.
Percent Calculations Every Day
Sales tax, tips, test grades—percent problems are practical and appear in everyday life constantly.
Summary Table: Problems and Answers
| Problem Type | Example Problem | Final Answer |
|---|---|---|
| Factoring Quadratics | x2+7x+10x2+7x+10 | (x+2)(x+5)(x+2)(x+5) |
| Solving Quadratic Equations | x2−9=0x2−9=0 | x=3,x=−3x=3,x=−3 |
| Percent Problem | What is 45% of 320? | 144 |
The specific subject of this practice can vary depending on your textbook. Search results indicate this page number and section title may be associated with:
- Go Math! 6th Grade: Solving percent problems.
- AP Biology: Data analysis and graphing.
- AP Physics 1: Force, motion, and foundational physics principles.
To ensure you have the right answers, match your textbook and problem set to the correct section below.
8.3 Independent Practice Page 221 Answer Key (Go Math! 6th Grade)
- Problem 1: [Answer and brief explanation]
- Problem 2: [Answer and brief explanation]
- Problem 3: [Answer and brief explanation]
- …and so on.
8.3 Independent Practice Page 221 Answer Key (AP Biology)
- Part A, Question i: [Answer and brief explanation]
- Part A, Question ii: [Answer and brief explanation]
- Part B: [Answer and brief explanation, potentially including a detailed image or graph if relevant]
- …and so on.
8.3 Independent Practice Page 221 Answer Key (AP Physics 1)
- Question 1: [Answer and brief explanation, including relevant formulas like Newton’s second law, if applicable]
- Question 2: [Answer and brief explanation]
- Question 3: [Answer and brief explanation]
- …and so on.
How to Use This Answer Key Effectively
This answer key is a tool for learning, not just a shortcut. To get the most out of it:
- Try to solve the problems first: Work through the independent practice problems on your own before checking the solutions.
- Identify your mistakes: Compare your answers to the key. If they are different, review the provided steps to understand where you went wrong.
- Learn the concepts: Don’t just copy the answers. Focus on understanding the underlying principles behind each solution.
- Use it as a study aid: This key can be a great way to review for a test or quiz by re-working problems you found challenging.
By following this approach, you’ll not only get the correct answers but also build a stronger foundation in the subject.