The Formula for Expanding a Quadratic Expression

Oct 13, 2025 by talynt.org 112 views

Introduction

In mathematics, the expanded form of an algebraic expression is a way of expressing it as a sum of simpler terms. This is particularly useful when dealing with quadratic expressions, which are expressions of the form $(x + a)^2$ . In this article, we will explore how to rewrite the expression $\left(x+\frac{1}{4}\right)^2$ in its expanded form.

The Formula for Expanding a Quadratic Expression

The formula for expanding a quadratic expression of the form $(x + a)^2$ is given by:

(x + a)^2 = x^2 + 2ax + a^2

This formula can be used to expand any quadratic expression of the form $(x + a)^2$ .

Applying the Formula to the Given Expression

Now, let's apply the formula to the given expression $\left(x+\frac{1}{4}\right)^2$ . We have:

\left(x+\frac{1}{4}\right)^2 = x^2 + 2x\left(\frac{1}{4}\right) + \left(\frac{1}{4}\right)^2

Using the formula, we can simplify this expression as follows:

\left(x+\frac{1}{4}\right)^2 = x^2 + \frac{1}{2}x + \frac{1}{16}

Step-by-Step Solution

Here's a step-by-step solution to the problem:

Apply the formula: Apply the formula for expanding a quadratic expression to the given expression $\left(x+\frac{1}{4}\right)^2$ .
Simplify the expression: Simplify the resulting expression by combining like terms.
Write the final answer: Write the final answer in the form $x^2 + bx + c$ .

Example Solution

Let's consider an example to illustrate the solution. Suppose we want to expand the expression $\left(x+\frac{1}{3}\right)^2$ . We can apply the formula as follows:

\left(x+\frac{1}{3}\right)^2 = x^2 + 2x\left(\frac{1}{3}\right) + \left(\frac{1}{3}\right)^2

Simplifying this expression, we get:

\left(x+\frac{1}{3}\right)^2 = x^2 + \frac{2}{3}x + \frac{1}{9}

Conclusion

In this article, we have explored how to rewrite the expression $\left(x+\frac{1}{4}\right)^2$ in its expanded form. We have applied the formula for expanding a quadratic expression and simplified the resulting expression to obtain the final answer. This technique can be used to expand any quadratic expression of the form $(x + a)^2$ .

Key Takeaways

The formula for expanding a quadratic expression is given by $(x + a)^2 = x^2 + 2ax + a^2$ .
To expand a quadratic expression, apply the formula and simplify the resulting expression by combining like terms.
The expanded form of a quadratic expression can be written in the form $x^2 + bx + c$ .

Glossary

Quadratic expression: An expression of the form $(x + a)^2$ .
Expanded form: The form of an expression as a sum of simpler terms.
Like terms: Terms that have the same variable and exponent.

References

Algebra
Mathematics
Quadratic Formula
Q&A: Expanding Quadratic Expressions =====================================

Introduction

In our previous article, we explored how to rewrite the expression $\left(x+\frac{1}{4}\right)^2$ in its expanded form. In this article, we will answer some frequently asked questions about expanding quadratic expressions.

Q: What is a quadratic expression?

A: A quadratic expression is an expression of the form $(x + a)^2$ , where $x$ is a variable and $a$ is a constant.

Q: How do I expand a quadratic expression?

A: To expand a quadratic expression, apply the formula $(x + a)^2 = x^2 + 2ax + a^2$ and simplify the resulting expression by combining like terms.

Q: What is the expanded form of a quadratic expression?

A: The expanded form of a quadratic expression is the form $x^2 + bx + c$ , where $b$ and $c$ are constants.

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, combine like terms by adding or subtracting the coefficients of the same variable.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable and exponent. For example, $2x^2$ and $3x^2$ are like terms because they have the same variable ( $x$ ) and exponent (2).

Q: Can I expand a quadratic expression with a negative coefficient?

A: Yes, you can expand a quadratic expression with a negative coefficient. For example, $(x - 2)^2 = x^2 - 4x + 4$ .

Q: Can I expand a quadratic expression with a fraction coefficient?

A: Yes, you can expand a quadratic expression with a fraction coefficient. For example, $(x + \frac{1}{2})^2 = x^2 + x + \frac{1}{4}$ .

Q: How do I apply the formula for expanding a quadratic expression?

A: To apply the formula for expanding a quadratic expression, substitute the values of $x$ and $a$ into the formula $(x + a)^2 = x^2 + 2ax + a^2$ and simplify the resulting expression.

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

A: Some common mistakes to avoid when expanding quadratic expressions include:

Forgetting to distribute the coefficient to the terms inside the parentheses
Not combining like terms correctly
Not simplifying the expression correctly

Conclusion

In this article, we have answered some frequently asked questions about expanding quadratic expressions. We hope this article has been helpful in clarifying any confusion you may have had about expanding quadratic expressions.

Key Takeaways

A quadratic expression is an expression of the form $(x + a)^2$ .
To expand a quadratic expression, apply the formula $(x + a)^2 = x^2 + 2ax + a^2$ and simplify the resulting expression by combining like terms.
Like terms are terms that have the same variable and exponent.
To simplify a quadratic expression, combine like terms by adding or subtracting the coefficients of the same variable.

Glossary

Quadratic expression: An expression of the form $(x + a)^2$ .
Expanded form: The form of an expression as a sum of simpler terms.
Like terms: Terms that have the same variable and exponent.