Which Will Result In A Difference Of Squares?A. $(-7x + 4)(-7x + 4$\] B. $(-7x + 4)(4 - 7x$\] C. $(-7x + 4)(-7x - 4$\] D. $(-7x + 4)(7x - 4$\]
In algebra, a difference of squares is a mathematical expression that can be factored into the product of two binomials. It is a fundamental concept in mathematics and is used to simplify complex expressions. In this article, we will explore which of the given options will result in a difference of squares.
What is a Difference of Squares?
A difference of squares is a mathematical expression of the form:
a^2 - b^2 = (a + b)(a - b)
where a and b are any two numbers or variables. This expression can be factored into the product of two binomials, (a + b) and (a - b).
How to Identify a Difference of Squares
To identify a difference of squares, we need to look for an expression that can be written in the form a^2 - b^2. We can do this by checking if the expression can be factored into the product of two binomials.
Option A:
Option A is a product of two identical binomials. When we multiply two identical binomials, we get a perfect square trinomial, not a difference of squares.
Option B:
Option B is a product of two binomials. When we multiply these binomials, we get:
(-7x + 4)(4 - 7x) = -28x^2 + 28x + 16 - 28x = -28x^2 - 12x + 16
This expression is not a difference of squares.
Option C:
Option C is a product of two binomials. When we multiply these binomials, we get:
(-7x + 4)(-7x - 4) = 49x^2 + 28x - 28x - 16 = 49x^2 - 16
This expression is a difference of squares.
Option D:
Option D is a product of two binomials. When we multiply these binomials, we get:
(-7x + 4)(7x - 4) = -49x^2 + 28x + 28x - 16 = -49x^2 - 16
This expression is not a difference of squares.
Conclusion
In conclusion, only Option C, , will result in a difference of squares. This is because the product of the two binomials can be factored into the product of two binomials, which is a characteristic of a difference of squares.
Key Takeaways
- A difference of squares is a mathematical expression that can be factored into the product of two binomials.
- To identify a difference of squares, we need to look for an expression that can be written in the form a^2 - b^2.
- Option C, , is the only option that will result in a difference of squares.
Practice Problems
- Factor the expression 9x^2 - 16 into the product of two binomials.
- Identify the difference of squares in the expression 25x^2 - 9.
- Simplify the expression (-7x + 4)(-7x - 4) by factoring it into the product of two binomials.
Answer Key
- (3x + 4)(3x - 4)
- 25x^2 - 9 = (5x + 3)(5x - 3)
- (-7x + 4)(-7x - 4) = 49x^2 - 16
Q&A: Difference of Squares =============================
In our previous article, we explored which of the given options will result in a difference of squares. In this article, we will answer some frequently asked questions about difference of squares.
Q: What is a difference of squares?
A: A difference of squares is a mathematical expression that can be factored into the product of two binomials. It is a fundamental concept in mathematics and is used to simplify complex expressions.
Q: How to identify a difference of squares?
A: To identify a difference of squares, we need to look for an expression that can be written in the form a^2 - b^2. We can do this by checking if the expression can be factored into the product of two binomials.
Q: What is the formula for a difference of squares?
A: The formula for a difference of squares is:
a^2 - b^2 = (a + b)(a - b)
where a and b are any two numbers or variables.
Q: Can a difference of squares be factored into the product of two binomials?
A: Yes, a difference of squares can be factored into the product of two binomials. This is the characteristic of a difference of squares.
Q: How to factor a difference of squares?
A: To factor a difference of squares, we need to identify the values of a and b in the expression a^2 - b^2. Then, we can factor the expression into the product of two binomials using the formula:
a^2 - b^2 = (a + b)(a - b)
Q: What is the difference between a difference of squares and a perfect square trinomial?
A: A difference of squares is a mathematical expression that can be factored into the product of two binomials, while a perfect square trinomial is a mathematical expression that can be factored into the square of a binomial.
Q: Can a difference of squares be used to simplify complex expressions?
A: Yes, a difference of squares can be used to simplify complex expressions. By factoring a difference of squares into the product of two binomials, we can simplify the expression and make it easier to work with.
Q: What are some common examples of difference of squares?
A: Some common examples of difference of squares include:
- 9x^2 - 16 = (3x + 4)(3x - 4)
- 25x^2 - 9 = (5x + 3)(5x - 3)
- 49x^2 - 16 = (7x + 4)(7x - 4)
Q: How to use difference of squares in real-world applications?
A: Difference of squares can be used in a variety of real-world applications, including:
- Simplifying complex expressions in algebra and calculus
- Factoring polynomials in mathematics and engineering
- Solving equations and inequalities in physics and engineering
Conclusion
In conclusion, difference of squares is a fundamental concept in mathematics that can be used to simplify complex expressions and factor polynomials. By understanding the formula and characteristics of a difference of squares, we can use it to solve a variety of problems in mathematics and real-world applications.
Key Takeaways
- A difference of squares is a mathematical expression that can be factored into the product of two binomials.
- To identify a difference of squares, we need to look for an expression that can be written in the form a^2 - b^2.
- A difference of squares can be used to simplify complex expressions and factor polynomials.
- Difference of squares can be used in a variety of real-world applications, including algebra, calculus, and physics.