Simplify The Expression: $\[ 5x^2 - 26x + 5 \\]

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying the given expression: $5x^2 - 26x + 5$. We will break down the process into manageable steps, making it easy to understand and follow along.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what we're dealing with. The given expression is a quadratic expression, which means it has a squared variable ($x^2$) and a linear term ($-26x$). The constant term is $5$. Our goal is to simplify this expression by combining like terms and factoring out any common factors.

Step 1: Factor Out Common Factors

The first step in simplifying the expression is to factor out any common factors. In this case, we can factor out a common factor of $5$ from all three terms:

$$5x^2 - 26x + 5 = 5(x^2 - 5.2x + 1)$$

By factoring out $5$, we have simplified the expression slightly, but we still have a quadratic expression inside the parentheses.

Step 2: Simplify the Quadratic Expression

Now that we have factored out $5$, we can focus on simplifying the quadratic expression inside the parentheses. To do this, we need to find two numbers whose product is $-5.2$ (the coefficient of the $x$ term) and whose sum is $0$ (the coefficient of the constant term). These numbers are $-2.6$ and $2.6$. We can rewrite the quadratic expression as:

$$x^2 - 5.2x + 1 = (x - 2.6)^2 - 1$$

By rewriting the quadratic expression in this form, we have simplified it further.

Step 3: Simplify the Expression

Now that we have simplified the quadratic expression, we can simplify the entire expression by combining the factored term with the simplified quadratic expression:

$$5(x^2 - 5.2x + 1) = 5((x - 2.6)^2 - 1)$$

By combining the factored term with the simplified quadratic expression, we have finally simplified the expression.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, we have simplified the given expression: $5x^2 - 26x + 5$. We factored out common factors, simplified the quadratic expression, and combined the factored term with the simplified quadratic expression to arrive at the final simplified expression. With practice and patience, you can master the art of simplifying algebraic expressions.

Tips and Tricks

• Always factor out common factors before simplifying the expression.
• Use the quadratic formula to find the roots of the quadratic expression.
• Simplify the expression by combining like terms and factoring out common factors.
• Use algebraic identities to simplify the expression.

Common Algebraic Identities

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

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. For example, in physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and optimize systems. In economics, we use algebraic expressions to model and analyze economic systems.

Conclusion

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Introduction

In our previous article, we explored the process of simplifying algebraic expressions. We broke down the steps involved in simplifying the expression $5x^2 - 26x + 5$. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

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

A: The first step in simplifying an algebraic expression is to factor out any common factors. This involves identifying any common factors that can be factored out of all the terms in the expression.

Q: How do I identify common factors?

A: To identify common factors, look for any numbers or variables that appear in all the terms of the expression. For example, in the expression $5x^2 - 26x + 5$, the common factor is $5$.

Q: What is the difference between factoring and simplifying an algebraic expression?

A: Factoring an algebraic expression involves breaking it down into its component parts, while simplifying an algebraic expression involves combining like terms and eliminating any unnecessary factors.

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, you can use the quadratic formula to find the roots of the expression. Alternatively, you can try to factor the expression by finding two numbers whose product is the constant term and whose sum is the coefficient of the $x$ term.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the roots of a quadratic equation. The formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: How do I use the quadratic formula to simplify a quadratic expression?

A: To use the quadratic formula to simplify a quadratic expression, you need to identify the values of $a$, $b$, and $c$ in the expression. Then, you can plug these values into the quadratic formula to find the roots of the expression.

Q: What are some common algebraic identities that I can use to simplify expressions?

A: Some common algebraic identities that you can use to simplify expressions include:

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

Q: How do I know when an expression is fully simplified?

A: An expression is fully simplified when there are no like terms left to combine and no unnecessary factors left to eliminate.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has numerous real-world applications, including:

• Physics: Algebraic expressions are used to describe the motion of objects.
• Engineering: Algebraic expressions are used to design and optimize systems.
• Economics: Algebraic expressions are used to model and analyze economic systems.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By following the steps outlined in this article, you can master the art of simplifying algebraic expressions. Remember to factor out common factors, simplify quadratic expressions, and use algebraic identities to simplify expressions. With practice and patience, you can become proficient in simplifying algebraic expressions.

Tips and Tricks

• Always factor out common factors before simplifying the expression.
• Use the quadratic formula to find the roots of the quadratic expression.
• Simplify the expression by combining like terms and factoring out common factors.
• Use algebraic identities to simplify the expression.

Common Algebraic Identities

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

Real-World Applications

• Physics: Algebraic expressions are used to describe the motion of objects.
• Engineering: Algebraic expressions are used to design and optimize systems.
• Economics: Algebraic expressions are used to model and analyze economic systems.