Simplify The Expression:${ \begin{array}{l} (5y+1)^2 - 4(y-1)^2 \ = [5y+1 - 4(y-1)][5y+1 + 4(y-1)] \ = (5y+1 - 4y + 4)(5y+1 + 4y - 4) \ = (y + 5)(9y + 5) \end{array} }$

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying a specific algebraic expression, which involves expanding and factoring quadratic expressions. We will break down the process into manageable steps, making it easier to understand and apply.

The Expression to Simplify

The given expression is:

{ \begin{array}{l} (5y+1)^2 - 4(y-1)^2 \\ = [5y+1 - 4(y-1)][5y+1 + 4(y-1)] \\ = (5y+1 - 4y + 4)(5y+1 + 4y - 4) \\ = (y + 5)(9y + 5) \end{array} \}

Step 1: Expand the Squared Terms

To simplify the expression, we start by expanding the squared terms using the formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2. In this case, we have:

{ (5y+1)^2 = (5y)^2 + 2(5y)(1) + 1^2 = 25y^2 + 10y + 1 \}

Step 2: Expand the Second Squared Term

Similarly, we expand the second squared term:

{ -4(y-1)^2 = -4(y^2 - 2y + 1) = -4y^2 + 8y - 4 \}

Step 3: Combine Like Terms

Now, we combine the expanded terms:

{ (5y+1)^2 - 4(y-1)^2 = 25y^2 + 10y + 1 - 4y^2 + 8y - 4 = 21y^2 + 18y - 3 \}

Step 4: Factor the Quadratic Expression

The next step is to factor the quadratic expression. We look for two numbers whose product is 21y2⋅(−3)21y^2 \cdot (-3) and whose sum is 18y18y. These numbers are yy and 9y9y, so we can write:

{ 21y^2 + 18y - 3 = (y + 5)(9y + 5) \}

Conclusion

In this article, we simplified the given algebraic expression by expanding and factoring quadratic expressions. We broke down the process into manageable steps, making it easier to understand and apply. By following these steps, you can simplify complex algebraic expressions and develop a deeper understanding of mathematical concepts.

Tips and Tricks

  • When simplifying algebraic expressions, always start by expanding the squared terms.
  • Use the formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 to expand squared terms.
  • Combine like terms to simplify the expression.
  • Factor the quadratic expression by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, algebraic expressions are used to describe the motion of objects, while in engineering, they are used to design and optimize systems. In economics, algebraic expressions are used to model economic systems and make predictions about future trends.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes to avoid:

  • Not expanding squared terms correctly.
  • Not combining like terms.
  • Not factoring the quadratic expression correctly.
  • Not checking the final answer for errors.

Final Thoughts

Simplifying algebraic expressions is an essential skill for students and professionals alike. By following the steps outlined in this article, you can simplify complex algebraic expressions and develop a deeper understanding of mathematical concepts. Remember to always start by expanding squared terms, combine like terms, and factor the quadratic expression correctly. With practice and patience, you can become proficient in simplifying algebraic expressions and apply this skill to real-world problems.

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Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to expand any squared terms using the formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2.

Q: How do I expand a squared term?

A: To expand a squared term, you need to multiply the two binomials together using the FOIL method (First, Outer, Inner, Last). For example, to expand (5y+1)2(5y+1)^2, you would multiply the two binomials together as follows:

{ (5y+1)^2 = (5y)(5y) + (5y)(1) + (1)(5y) + (1)(1) = 25y^2 + 10y + 10y + 1 = 25y^2 + 20y + 1 \}

Q: What is the difference between expanding and factoring?

A: Expanding an algebraic expression involves multiplying out the terms to get a simpler expression, while factoring involves breaking down an expression into simpler terms.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, to factor the quadratic expression x2+5x+6x^2 + 5x + 6, you would look for two numbers whose product is 66 and whose sum is 55. These numbers are 22 and 33, so you can write:

{ x^2 + 5x + 6 = (x + 2)(x + 3) \}

Q: What is the difference between a monomial and a binomial?

A: A monomial is a single term that consists of a number or a variable, while a binomial is a sum of two terms.

Q: How do I simplify an algebraic expression with multiple terms?

A: To simplify an algebraic expression with multiple terms, you need to combine like terms by adding or subtracting the coefficients of the same variables.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to check the final answer for errors and make sure that it is in the simplest form possible.

Q: Can I use a calculator to simplify algebraic expressions?

A: Yes, you can use a calculator to simplify algebraic expressions, but it's always a good idea to check the final answer by hand to make sure that it is correct.

Q: How do I know if an algebraic expression is in its simplest form?

A: An algebraic expression is in its simplest form if it cannot be simplified any further by combining like terms or factoring.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

  • Not expanding squared terms correctly.
  • Not combining like terms.
  • Not factoring the quadratic expression correctly.
  • Not checking the final answer for errors.

Q: Can I use algebraic expressions to solve real-world problems?

A: Yes, algebraic expressions can be used to solve real-world problems in fields such as physics, engineering, and economics.

Q: How do I apply algebraic expressions to real-world problems?

A: To apply algebraic expressions to real-world problems, you need to identify the variables and constants in the problem, set up an equation using the given information, and then solve for the unknown variable.

Q: What are some examples of real-world problems that involve algebraic expressions?

A: Some examples of real-world problems that involve algebraic expressions include:

  • Modeling the motion of an object using algebraic expressions.
  • Designing and optimizing systems using algebraic expressions.
  • Analyzing economic systems using algebraic expressions.

Q: Can I use algebraic expressions to solve problems in other areas of mathematics?

A: Yes, algebraic expressions can be used to solve problems in other areas of mathematics, such as geometry and trigonometry.

Q: How do I apply algebraic expressions to other areas of mathematics?

A: To apply algebraic expressions to other areas of mathematics, you need to identify the variables and constants in the problem, set up an equation using the given information, and then solve for the unknown variable.

Q: What are some examples of problems in other areas of mathematics that involve algebraic expressions?

A: Some examples of problems in other areas of mathematics that involve algebraic expressions include:

  • Solving geometric problems using algebraic expressions.
  • Analyzing trigonometric functions using algebraic expressions.

Q: Can I use algebraic expressions to solve problems in science and engineering?

A: Yes, algebraic expressions can be used to solve problems in science and engineering.

Q: How do I apply algebraic expressions to science and engineering problems?

A: To apply algebraic expressions to science and engineering problems, you need to identify the variables and constants in the problem, set up an equation using the given information, and then solve for the unknown variable.

Q: What are some examples of problems in science and engineering that involve algebraic expressions?

A: Some examples of problems in science and engineering that involve algebraic expressions include:

  • Modeling the motion of an object using algebraic expressions.
  • Designing and optimizing systems using algebraic expressions.
  • Analyzing economic systems using algebraic expressions.

Q: Can I use algebraic expressions to solve problems in economics?

A: Yes, algebraic expressions can be used to solve problems in economics.

Q: How do I apply algebraic expressions to economic problems?

A: To apply algebraic expressions to economic problems, you need to identify the variables and constants in the problem, set up an equation using the given information, and then solve for the unknown variable.

Q: What are some examples of problems in economics that involve algebraic expressions?

A: Some examples of problems in economics that involve algebraic expressions include:

  • Analyzing economic systems using algebraic expressions.
  • Modeling the behavior of economic variables using algebraic expressions.
  • Solving optimization problems using algebraic expressions.