Examples of Difference of Squares

Oct 13, 2025 by talynt.org 92 views

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 shows a difference of squares.

What is a Difference of Squares?

A difference of squares is a mathematical expression of the form $a^2 - b^2$ , where $a$ and $b$ are algebraic expressions. This expression can be factored into the product of two binomials: $(a+b)(a-b)$ . For example, the expression $9x^2 - 16y^2$ can be factored as $(3x+4y)(3x-4y)$ .

Option A: $10y^2 - 4x^2$

Let's analyze the first option: $10y^2 - 4x^2$ . This expression can be rewritten as $(\sqrt{10}y)^2 - (\sqrt{4}x)^2$ . Since $\sqrt{10}$ and $\sqrt{4}$ are not perfect squares, this expression does not show a difference of squares.

Option B: $16y^2 - x^2$

Now, let's analyze the second option: $16y^2 - x^2$ . This expression can be rewritten as $(4y)^2 - (x)^2$ . Since $4y$ and $x$ are not perfect squares, this expression does not show a difference of squares.

Option C: $8x^2 - 40x + 25$

Next, let's analyze the third option: $8x^2 - 40x + 25$ . This expression can be rewritten as $(2x)^2 - 2(2x)(5) + (5)^2$ . This expression is a perfect square trinomial, but it does not show a difference of squares.

Option D: $64x^2 - 48x + 9$

Finally, let's analyze the fourth option: $64x^2 - 48x + 9$ . This expression can be rewritten as $(8x)^2 - 2(8x)(\frac{3}{2}) + (\frac{3}{2})^2$ . This expression is also a perfect square trinomial, but it does not show a difference of squares.

Conclusion

In conclusion, none of the given options show a difference of squares. A difference of squares is a mathematical expression of the form $a^2 - b^2$ , where $a$ and $b$ are algebraic expressions. The expressions in the given options do not meet this criteria, and therefore, do not show a difference of squares.

What is a Perfect Square Trinomial?

A perfect square trinomial is a quadratic expression of the form $a^2 - 2ab + b^2$ , where $a$ and $b$ are algebraic expressions. This expression can be factored into the product of two binomials: $(a-b)^2$ . For example, the expression $9x^2 - 12x + 4$ can be factored as $(3x-2)^2$ .

How to Identify a Difference of Squares?

To identify a difference of squares, look for an expression of the form $a^2 - b^2$ , where $a$ and $b$ are algebraic expressions. If the expression can be rewritten in this form, then it shows a difference of squares. For example, the expression $25x^2 - 9y^2$ can be rewritten as $(5x)^2 - (3y)^2$ , which shows a difference of squares.

Examples of Difference of Squares

Here are some examples of difference of squares:

$9x^2 - 16y^2 = (3x+4y)(3x-4y)$
$25x^2 - 9y^2 = (5x+3y)(5x-3y)$
$36x^2 - 49y^2 = (6x+7y)(6x-7y)$

Conclusion

In our previous article, we discussed what a difference of squares is and how to identify it. 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 of the form $a^2 - b^2$ , where $a$ and $b$ are algebraic expressions. This expression can be factored into the product of two binomials: $(a+b)(a-b)$ .

Q: How do I identify a difference of squares?

A: To identify a difference of squares, look for an expression of the form $a^2 - b^2$ , where $a$ and $b$ are algebraic expressions. If the expression can be rewritten in this form, then it shows a difference of squares.

Q: What are some examples of difference of squares?

A: Here are some examples of difference of squares:

$9x^2 - 16y^2 = (3x+4y)(3x-4y)$
$25x^2 - 9y^2 = (5x+3y)(5x-3y)$
$36x^2 - 49y^2 = (6x+7y)(6x-7y)$

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: $(a+b)(a-b)$ .

Q: What is the difference between a difference of squares and a perfect square trinomial?

A: A difference of squares is a mathematical expression of the form $a^2 - b^2$ , where $a$ and $b$ are algebraic expressions. A perfect square trinomial is a quadratic expression of the form $a^2 - 2ab + b^2$ , where $a$ and $b$ are algebraic expressions.

Q: Can a perfect square trinomial be factored into the product of two binomials?

A: Yes, a perfect square trinomial can be factored into the product of two binomials: $(a-b)^2$ .

Q: How do I factor a difference of squares?

A: To factor a difference of squares, follow these steps:

Identify the expression as a difference of squares.
Rewrite the expression in the form $a^2 - b^2$ .
Factor the expression into the product of two binomials: $(a+b)(a-b)$ .

Q: What are some common mistakes to avoid when factoring a difference of squares?

A: Here are some common mistakes to avoid when factoring a difference of squares:

Not identifying the expression as a difference of squares.
Not rewriting the expression in the form $a^2 - b^2$ .
Not factoring the expression into the product of two binomials: $(a+b)(a-b)$ .

Conclusion

In conclusion, a difference of squares is a mathematical expression of the form $a^2 - b^2$ , where $a$ and $b$ are algebraic expressions. We hope this article has helped you understand what a difference of squares is and how to identify it. Remember to avoid common mistakes when factoring a difference of squares.