Simplify The Expression: ${ \frac{x^2 - 9x + 14}{5x^2 - 5x - 10} }$

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Introduction

Simplifying rational expressions is a crucial skill in algebra, and it requires a deep understanding of factoring, fractions, and algebraic manipulation. In this article, we will guide you through the process of simplifying the given expression: ${ \frac{x^2 - 9x + 14}{5x^2 - 5x - 10} }$

Understanding the Expression

The given expression is a rational expression, which is a fraction that contains variables and constants in the numerator and denominator. To simplify this expression, we need to factor both the numerator and the denominator.

Factoring the Numerator

The numerator of the expression is x2−9x+14x^2 - 9x + 14. To factor this quadratic expression, we need to find two numbers whose product is 14 and whose sum is -9. These numbers are -7 and -2, so we can write the numerator as:

x2−9x+14=(x−7)(x−2)x^2 - 9x + 14 = (x - 7)(x - 2)

Factoring the Denominator

The denominator of the expression is 5x2−5x−105x^2 - 5x - 10. To factor this quadratic expression, we need to find two numbers whose product is -50 and whose sum is -5. These numbers are -10 and 5, so we can write the denominator as:

5x2−5x−10=5(x2−x−2)5x^2 - 5x - 10 = 5(x^2 - x - 2)

We can further factor the quadratic expression inside the parentheses as:

5(x2−x−2)=5(x−2)(x+1)5(x^2 - x - 2) = 5(x - 2)(x + 1)

Simplifying the Expression

Now that we have factored both the numerator and the denominator, we can simplify the expression by canceling out any common factors.

{ \frac{(x - 7)(x - 2)}{5(x - 2)(x + 1)} \}

We can see that the factor (x−2)(x - 2) appears in both the numerator and the denominator, so we can cancel it out:

{ \frac{(x - 7)}{5(x + 1)} \}

This is the simplified form of the expression.

Conclusion

Simplifying rational expressions requires a deep understanding of factoring, fractions, and algebraic manipulation. By following the steps outlined in this article, you can simplify even the most complex rational expressions. Remember to always factor both the numerator and the denominator, and then cancel out any common factors to simplify the expression.

Tips and Tricks

  • Always factor both the numerator and the denominator before simplifying the expression.
  • Look for common factors between the numerator and the denominator, and cancel them out.
  • Use algebraic manipulation to simplify the expression, such as multiplying or dividing both the numerator and the denominator by the same value.

Real-World Applications

Simplifying rational expressions has many real-world applications, such as:

  • Science and Engineering: Simplifying rational expressions is crucial in science and engineering, where complex equations need to be solved to model real-world phenomena.
  • Finance: Simplifying rational expressions is used in finance to calculate interest rates, investment returns, and other financial metrics.
  • Computer Science: Simplifying rational expressions is used in computer science to optimize algorithms and solve complex problems.

Common Mistakes to Avoid

  • Not factoring the numerator and denominator: Failing to factor both the numerator and the denominator can lead to incorrect simplifications.
  • Not canceling out common factors: Failing to cancel out common factors can lead to incorrect simplifications.
  • Not using algebraic manipulation: Failing to use algebraic manipulation can lead to incorrect simplifications.

Final Thoughts

Simplifying rational expressions is a crucial skill in algebra, and it requires a deep understanding of factoring, fractions, and algebraic manipulation. By following the steps outlined in this article, you can simplify even the most complex rational expressions. Remember to always factor both the numerator and the denominator, and then cancel out any common factors to simplify the expression.

Introduction

In our previous article, we guided you through the process of simplifying the expression: ${ \frac{x^2 - 9x + 14}{5x^2 - 5x - 10} }$

In this article, we will answer some of the most frequently asked questions about simplifying rational expressions.

Q&A

Q: What is the first step in simplifying a rational expression?

A: The first step in simplifying a rational expression is to factor both the numerator and the denominator.

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 middle term. You can then write the quadratic expression as the product of two binomials.

Q: What is the difference between factoring and simplifying a rational expression?

A: Factoring a rational expression involves breaking it down into its simplest form by expressing it as the product of two or more factors. Simplifying a rational expression involves canceling out any common factors between the numerator and the denominator.

Q: Can I simplify a rational expression if the numerator and denominator have no common factors?

A: Yes, you can still simplify a rational expression even if the numerator and denominator have no common factors. In this case, the expression is already in its simplest form.

Q: How do I know if a rational expression is in its simplest form?

A: A rational expression is in its simplest form if there are no common factors between the numerator and the denominator.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you need to be careful not to divide by zero.

Q: What is the rule for canceling out common factors in a rational expression?

A: The rule for canceling out common factors in a rational expression is to cancel out any common factors between the numerator and the denominator.

Q: Can I cancel out a common factor if it is raised to a power?

A: Yes, you can cancel out a common factor if it is raised to a power. For example, if you have the expression ${ \frac{x^2 - 9x + 14}{5x^2 - 5x - 10} }$, you can cancel out the factor (x−2)(x - 2) even if it is raised to a power.

Q: What is the final step in simplifying a rational expression?

A: The final step in simplifying a rational expression is to write the expression in its simplest form.

Tips and Tricks

  • Always factor both the numerator and the denominator before simplifying the expression.
  • Look for common factors between the numerator and the denominator, and cancel them out.
  • Use algebraic manipulation to simplify the expression, such as multiplying or dividing both the numerator and the denominator by the same value.
  • Be careful not to divide by zero when simplifying a rational expression with a variable in the denominator.

Common Mistakes to Avoid

  • Not factoring the numerator and denominator: Failing to factor both the numerator and the denominator can lead to incorrect simplifications.
  • Not canceling out common factors: Failing to cancel out common factors can lead to incorrect simplifications.
  • Not using algebraic manipulation: Failing to use algebraic manipulation can lead to incorrect simplifications.
  • Dividing by zero: Dividing by zero is undefined and can lead to incorrect simplifications.

Final Thoughts

Simplifying rational expressions is a crucial skill in algebra, and it requires a deep understanding of factoring, fractions, and algebraic manipulation. By following the steps outlined in this article, you can simplify even the most complex rational expressions. Remember to always factor both the numerator and the denominator, and then cancel out any common factors to simplify the expression.