Factor The Expression: 9 X 2 − 49 Y 2 9x^2 - 49y^2 9 X 2 − 49 Y 2

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Introduction

In mathematics, factoring is a process of expressing an algebraic expression as a product of simpler expressions. It is an essential skill in algebra and is used to simplify complex expressions, solve equations, and graph functions. In this article, we will focus on factoring the expression $9x^2 - 49y^2$. We will use the difference of squares formula to factor this expression.

The Difference of Squares Formula

The difference of squares formula is a fundamental concept in algebra that states:

$a^2 - b^2 = (a + b)(a - b)$

This formula can be used to factor expressions of the form $a^2 - b^2$, where $a$ and $b$ are any real numbers.

Factoring the Expression

To factor the expression $9x^2 - 49y^2$, we can use the difference of squares formula. We can rewrite the expression as:

$9x^2 - 49y^2 = (3x)^2 - (7y)^2$

Now, we can apply the difference of squares formula:

$(3x)^2 - (7y)^2 = (3x + 7y)(3x - 7y)$

Therefore, the factored form of the expression $9x^2 - 49y^2$ is:

$(3x + 7y)(3x - 7y)$

Example

Let's consider an example to illustrate the concept of factoring the expression $9x^2 - 49y^2$. Suppose we want to factor the expression $9x^2 - 49y^2$ and then substitute $x = 2$ and $y = 3$ into the factored form.

First, we can factor the expression using the difference of squares formula:

$9x^2 - 49y^2 = (3x)^2 - (7y)^2 = (3x + 7y)(3x - 7y)$

Now, we can substitute $x = 2$ and $y = 3$ into the factored form:

$(3x + 7y)(3x - 7y) = (3(2) + 7(3))(3(2) - 7(3))$

$= (6 + 21)(6 - 21)$

$= (27)(-15)$

$= -405$

Therefore, the value of the expression $9x^2 - 49y^2$ when $x = 2$ and $y = 3$ is $-405$.

Conclusion

In this article, we have discussed how to factor the expression $9x^2 - 49y^2$ using the difference of squares formula. We have also provided an example to illustrate the concept of factoring the expression and then substituting values into the factored form. Factoring is an essential skill in algebra that can be used to simplify complex expressions, solve equations, and graph functions. By mastering the difference of squares formula, you can factor expressions of the form $a^2 - b^2$ and solve a wide range of algebraic problems.

Tips and Tricks

• To factor an expression of the form $a^2 - b^2$, use the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$.
• Make sure to identify the values of $a$ and $b$ in the expression before applying the difference of squares formula.
• When factoring an expression, look for common factors that can be factored out.
• Use the distributive property to expand the factored form and verify that it is equivalent to the original expression.

Common Mistakes

• Failing to identify the values of $a$ and $b$ in the expression before applying the difference of squares formula.
• Not using the distributive property to expand the factored form and verify that it is equivalent to the original expression.
• Not factoring out common factors that can be factored out.

Real-World Applications

Factoring is an essential skill in algebra that has numerous real-world applications. Some examples include:

• Simplifying complex expressions: Factoring can be used to simplify complex expressions and make them easier to work with.
• Solving equations: Factoring can be used to solve equations by factoring out common factors and then solving for the remaining variables.
• Graphing functions: Factoring can be used to graph functions by factoring out common factors and then using the factored form to graph the function.

Conclusion

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Q: What is the difference of squares formula?

A: The difference of squares formula is a fundamental concept in algebra that states:

$a^2 - b^2 = (a + b)(a - b)$

This formula can be used to factor expressions of the form $a^2 - b^2$, where $a$ and $b$ are any real numbers.

Q: How do I factor the expression $9x^2 - 49y^2$?

A: To factor the expression $9x^2 - 49y^2$, we can use the difference of squares formula. We can rewrite the expression as:

$9x^2 - 49y^2 = (3x)^2 - (7y)^2$

Now, we can apply the difference of squares formula:

$(3x)^2 - (7y)^2 = (3x + 7y)(3x - 7y)$

Therefore, the factored form of the expression $9x^2 - 49y^2$ is:

$(3x + 7y)(3x - 7y)$

Q: What if I have an expression of the form $a^2 + b^2$? Can I still use the difference of squares formula?

A: No, you cannot use the difference of squares formula to factor an expression of the form $a^2 + b^2$. The difference of squares formula only works for expressions of the form $a^2 - b^2$. If you have an expression of the form $a^2 + b^2$, you will need to use a different factoring technique.

Q: Can I factor an expression with a coefficient of 1?

A: Yes, you can factor an expression with a coefficient of 1. For example, the expression $x^2 - 4$ can be factored as:

$x^2 - 4 = (x + 2)(x - 2)$

Q: What if I have an expression with a coefficient of 0? Can I still factor it?

A: No, you cannot factor an expression with a coefficient of 0. For example, the expression $0x^2 - 49y^2$ cannot be factored because it is equal to 0.

Q: Can I factor an expression with a variable in the denominator?

A: No, you cannot factor an expression with a variable in the denominator. For example, the expression $\frac{x^2}{y^2} - 1$ cannot be factored because it has a variable in the denominator.

Q: What if I have an expression with a negative sign in front of it? Can I still factor it?

A: Yes, you can factor an expression with a negative sign in front of it. For example, the expression $-x^2 + 49y^2$ can be factored as:

$-x^2 + 49y^2 = -(x^2 - 49y^2) = -(x + 7y)(x - 7y)$

Q: Can I factor an expression with a fraction?

A: No, you cannot factor an expression with a fraction. For example, the expression $\frac{x^2}{y^2} - 1$ cannot be factored because it has a fraction.

Q: What if I have an expression with a square root in it? Can I still factor it?

A: No, you cannot factor an expression with a square root in it. For example, the expression $\sqrt{x^2} - 49y^2$ cannot be factored because it has a square root.

Conclusion

In this Q&A article, we have discussed some common questions and answers related to factoring the expression $9x^2 - 49y^2$. We have covered topics such as the difference of squares formula, factoring expressions with coefficients, and factoring expressions with variables in the denominator. By mastering the difference of squares formula and understanding these common questions and answers, you can factor expressions of the form $a^2 - b^2$ and solve a wide range of algebraic problems.