Simplify The Expression:$\frac{25x - 4x^2 - 25}{4x^2 + 15x - 25}$

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities more efficiently. It involves manipulating the given expression to make it easier to work with, often by combining like terms, factoring, or canceling out common factors. In this article, we will simplify the expression $\frac{25x - 4x^2 - 25}{4x^2 + 15x - 25}$ using various algebraic techniques.

Understanding the Expression

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Before we start simplifying the expression, let's take a closer look at it. The given expression is a rational function, which is a ratio of two polynomials. The numerator is $25x - 4x^2 - 25$, and the denominator is $4x^2 + 15x - 25$. Our goal is to simplify this expression by manipulating the numerator and denominator.

Factoring the Numerator and Denominator

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One of the most effective ways to simplify a rational function is to factor the numerator and denominator. Factoring involves expressing a polynomial as a product of simpler polynomials. Let's start by factoring the numerator and denominator.

Factoring the Numerator

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The numerator is $25x - 4x^2 - 25$. We can factor out a common factor of $-1$ from the first two terms:

$$25x - 4x^2 - 25 = -(4x^2 + 25x + 25)$$

However, this doesn't seem to help much. Let's try factoring the numerator in a different way. We can factor out a common factor of $-1$ from the first and last terms:

$$25x - 4x^2 - 25 = -(4x^2 + 25x) + 25$$

Now, we can factor the first two terms as a difference of squares:

$$-(4x^2 + 25x) = -(4x^2 - 25x + 25x) = -(4x^2 - 25x)(x + 1)$$

So, the numerator can be factored as:

$$25x - 4x^2 - 25 = -(4x^2 - 25x)(x + 1) + 25$$

Factoring the Denominator

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The denominator is $4x^2 + 15x - 25$. We can factor this quadratic expression as:

$$4x^2 + 15x - 25 = (4x - 5)(x + 5)$$

Canceling Out Common Factors

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Now that we have factored the numerator and denominator, we can cancel out common factors. The numerator is $-(4x^2 - 25x)(x + 1) + 25$, and the denominator is $(4x - 5)(x + 5)$. We can see that both the numerator and denominator have a common factor of $(x + 1)$.

By canceling out the common factor of $(x + 1)$, we get:

$$\frac{25x - 4x^2 - 25}{4x^2 + 15x - 25} = \frac{-(4x^2 - 25x)}{(4x - 5)(x + 5)} + \frac{25}{(4x - 5)(x + 5)}$$

Simplifying the Expression

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Now that we have canceled out the common factor of $(x + 1)$, we can simplify the expression further. We can combine the two fractions into a single fraction:

$$\frac{25x - 4x^2 - 25}{4x^2 + 15x - 25} = \frac{-(4x^2 - 25x) + 25}{(4x - 5)(x + 5)}$$

Final Simplification

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We can simplify the expression further by combining the numerator:

$$\frac{25x - 4x^2 - 25}{4x^2 + 15x - 25} = \frac{-(4x^2 - 25x) + 25}{(4x - 5)(x + 5)} = \frac{-4x^2 + 25x + 25}{(4x - 5)(x + 5)}$$

Conclusion

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In this article, we simplified the expression $\frac{25x - 4x^2 - 25}{4x^2 + 15x - 25}$ using various algebraic techniques. We factored the numerator and denominator, canceled out common factors, and simplified the expression further. The final simplified expression is $\frac{-4x^2 + 25x + 25}{(4x - 5)(x + 5)}$. This expression is much easier to work with than the original expression, and it can be used to solve equations and inequalities more efficiently.

Example Use Case

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The simplified expression $\frac{-4x^2 + 25x + 25}{(4x - 5)(x + 5)}$ can be used to solve equations and inequalities in various mathematical contexts. For example, suppose we want to find the values of $x$ that satisfy the equation $\frac{25x - 4x^2 - 25}{4x^2 + 15x - 25} = 0$. We can use the simplified expression to solve for $x$:

$$\frac{-4x^2 + 25x + 25}{(4x - 5)(x + 5)} = 0$$

This implies that the numerator $-4x^2 + 25x + 25$ must be equal to zero. We can solve for $x$ by factoring the numerator:

$$-4x^2 + 25x + 25 = -(4x^2 - 25x - 25) = -(4x - 5)(x + 5)$$

This implies that either $4x - 5 = 0$ or $x + 5 = 0$. Solving for $x$, we get:

$$x = \frac{5}{4} \text{ or } x = -5$$

Therefore, the values of $x$ that satisfy the equation are $x = \frac{5}{4}$ and $x = -5$.

Final Thoughts

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In conclusion, simplifying expressions is a crucial skill in algebra that helps us solve equations and inequalities more efficiently. By factoring the numerator and denominator, canceling out common factors, and simplifying the expression further, we can make the expression easier to work with. The simplified expression $\frac{-4x^2 + 25x + 25}{(4x - 5)(x + 5)}$ can be used to solve equations and inequalities in various mathematical contexts.

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Introduction

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In our previous article, we simplified the expression $\frac{25x - 4x^2 - 25}{4x^2 + 15x - 25}$ using various algebraic techniques. In this article, we will answer some common questions that students often have when simplifying expressions.

Q&A

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Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to factor the numerator and denominator. Factoring involves expressing a polynomial as a product of simpler polynomials.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you can use the factoring method, which involves finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the middle term.

Q: What is the difference between factoring and canceling out common factors?

A: Factoring involves expressing a polynomial as a product of simpler polynomials, while canceling out common factors involves eliminating common factors between the numerator and denominator.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to identify the common factors between the numerator and denominator and eliminate them.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{-4x^2 + 25x + 25}{(4x - 5)(x + 5)}$.

Q: How do I use the simplified expression to solve equations and inequalities?

A: You can use the simplified expression to solve equations and inequalities by setting the numerator equal to zero and solving for $x$.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not factoring the numerator and denominator
• Not canceling out common factors
• Not simplifying the expression further
• Not checking for extraneous solutions

Example Use Case

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Suppose we want to find the values of $x$ that satisfy the equation $\frac{25x - 4x^2 - 25}{4x^2 + 15x - 25} = 0$. We can use the simplified expression to solve for $x$:

$$\frac{-4x^2 + 25x + 25}{(4x - 5)(x + 5)} = 0$$

This implies that the numerator $-4x^2 + 25x + 25$ must be equal to zero. We can solve for $x$ by factoring the numerator:

$$-4x^2 + 25x + 25 = -(4x^2 - 25x - 25) = -(4x - 5)(x + 5)$$

This implies that either $4x - 5 = 0$ or $x + 5 = 0$. Solving for $x$, we get:

$$x = \frac{5}{4} \text{ or } x = -5$$

Therefore, the values of $x$ that satisfy the equation are $x = \frac{5}{4}$ and $x = -5$.

Final Thoughts

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In conclusion, simplifying expressions is a crucial skill in algebra that helps us solve equations and inequalities more efficiently. By factoring the numerator and denominator, canceling out common factors, and simplifying the expression further, we can make the expression easier to work with. The simplified expression $\frac{-4x^2 + 25x + 25}{(4x - 5)(x + 5)}$ can be used to solve equations and inequalities in various mathematical contexts.

Additional Resources

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For more information on simplifying expressions, you can refer to the following resources:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

Conclusion

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In this article, we answered some common questions that students often have when simplifying expressions. We also provided an example use case and additional resources for further learning. By mastering the skills of simplifying expressions, you can become more proficient in algebra and solve equations and inequalities more efficiently.