Simplify And Evaluate The Expression $\frac{a^2-9}{6a^2-18a}$ For $a = -0.3$.

Introduction

In this article, we will simplify and evaluate the expression $\frac{a^2-9}{6a^2-18a}$ for a given value of $a$. We will use algebraic techniques to simplify the expression and then substitute the value of $a$ to obtain the final result.

Step 1: Factor the Numerator and Denominator

The first step in simplifying the expression is to factor the numerator and denominator. The numerator can be factored as a difference of squares:

$$a^2-9 = (a-3)(a+3)$$

The denominator can be factored as a difference of squares:

$$6a^2-18a = 6a(a-3)$$

Step 2: Simplify the Expression

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

$$\frac{(a-3)(a+3)}{6a(a-3)} = \frac{a+3}{6a}$$

Step 3: Substitute the Value of $a$

Now that we have simplified the expression, we can substitute the value of $a$ to obtain the final result. We are given that $a = -0.3$, so we can substitute this value into the simplified expression:

$$\frac{a+3}{6a} = \frac{-0.3+3}{6(-0.3)} = \frac{2.7}{-1.8}$$

Step 4: Evaluate the Expression

Now that we have substituted the value of $a$, we can evaluate the expression to obtain the final result:

$$\frac{2.7}{-1.8} = -1.5$$

Conclusion

In this article, we simplified and evaluated the expression $\frac{a^2-9}{6a^2-18a}$ for $a = -0.3$. We used algebraic techniques to simplify the expression and then substituted the value of $a$ to obtain the final result. The final result is $-1.5$.

Mathematical Background

The expression $\frac{a^2-9}{6a^2-18a}$ is a rational expression, which is a fraction with a polynomial in the numerator and a polynomial in the denominator. To simplify a rational expression, we can factor the numerator and denominator and then cancel out any common factors.

Real-World Applications

Rational expressions have many real-world applications, including physics, engineering, and economics. For example, in physics, rational expressions can be used to model the motion of objects, while in engineering, they can be used to design electrical circuits. In economics, rational expressions can be used to model the behavior of markets.

Tips and Tricks

When simplifying rational expressions, it is often helpful to factor the numerator and denominator and then cancel out any common factors. Additionally, it is a good idea to check your work by plugging in a value for the variable and evaluating the expression.

Common Mistakes

When simplifying rational expressions, it is easy to make mistakes. Some common mistakes include:

• Not factoring the numerator and denominator
• Not canceling out common factors
• Not checking your work

Conclusion

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Introduction

In our previous article, we simplified and evaluated the expression $\frac{a^2-9}{6a^2-18a}$ for $a = -0.3$. In this article, we will answer some common questions that readers may have about simplifying and evaluating rational expressions.

Q: What is a rational expression?

A: A rational expression is a fraction with a polynomial in the numerator and a polynomial in the denominator. Rational expressions can be simplified by factoring the numerator and denominator and canceling out any common factors.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, follow these steps:

1. Factor the numerator and denominator.
2. Cancel out any common factors.
3. Check your work by plugging in a value for the variable and evaluating the expression.

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

A: A rational number is a number that can be expressed as the ratio of two integers, such as $\frac{1}{2}$ or $\frac{3}{4}$. A rational expression, on the other hand, is a fraction with a polynomial in the numerator and a polynomial in 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 must be careful not to divide by zero. If the variable is a factor of the denominator, you can cancel it out.

Q: How do I evaluate a rational expression with a variable in the denominator?

A: To evaluate a rational expression with a variable in the denominator, follow these steps:

1. Plug in the value of the variable into the expression.
2. Simplify the expression by canceling out any common factors.
3. Evaluate the expression to obtain the final result.

Q: What is the difference between a rational expression and an algebraic expression?

A: An algebraic expression is a mathematical expression that contains variables and constants, such as $2x + 3$ or $x^2 - 4$. A rational expression, on the other hand, is a fraction with a polynomial in the numerator and a polynomial in the denominator.

Q: Can I simplify a rational expression with a negative exponent?

A: Yes, you can simplify a rational expression with a negative exponent. To do this, follow these steps:

1. Rewrite the expression with a positive exponent.
2. Simplify the expression by canceling out any common factors.
3. Evaluate the expression to obtain the final result.

Q: How do I simplify a rational expression with a fraction in the numerator or denominator?

A: To simplify a rational expression with a fraction in the numerator or denominator, follow these steps:

1. Simplify the fraction in the numerator or denominator.
2. Factor the numerator and denominator.
3. Cancel out any common factors.
4. Evaluate the expression to obtain the final result.

Conclusion

In conclusion, simplifying and evaluating rational expressions requires the use of algebraic techniques, including factoring and canceling out common factors. By following these techniques, we can simplify rational expressions and obtain the final result.