Select The Expression That Is Equivalent To $(x+6)^2$.A. $2x + 12x + 12$B. \$x^2 + 12x^2 + 36$[/tex\]C. $x^2 + 6x + 36$D. $x^2 + 12x + 36$

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Introduction

Quadratic expressions are a fundamental concept in mathematics, and understanding how to solve them is crucial for success in algebra and beyond. In this article, we will explore the concept of quadratic expressions, specifically focusing on the expression (x+6)2(x+6)^2. We will examine the different options provided and determine which one is equivalent to the given expression.

What is a Quadratic Expression?

A quadratic expression is a polynomial expression of degree two, which means it has a variable raised to the power of two. The general form of a quadratic expression is ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants, and xx is the variable.

Expanding the Given Expression

To determine which option is equivalent to the given expression, we need to expand it using the formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2. In this case, a=xa = x and b=6b = 6. Therefore, we can expand the expression as follows:

(x+6)2=x2+2(x)(6)+62(x+6)^2 = x^2 + 2(x)(6) + 6^2

Simplifying the Expression

Now that we have expanded the expression, we can simplify it by evaluating the products and combining like terms.

x2+2(x)(6)+62=x2+12x+36x^2 + 2(x)(6) + 6^2 = x^2 + 12x + 36

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options provided to determine which one is equivalent.

  • Option A: 2x+12x+122x + 12x + 12
  • Option B: x2+12x2+36x^2 + 12x^2 + 36
  • Option C: x2+6x+36x^2 + 6x + 36
  • Option D: x2+12x+36x^2 + 12x + 36

Conclusion

Based on our analysis, we can conclude that the correct answer is Option D: x2+12x+36x^2 + 12x + 36. This is because it matches the simplified expression we obtained by expanding and simplifying the given expression.

Why is this Important?

Understanding how to solve quadratic expressions is crucial for success in mathematics and science. Quadratic expressions are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits. By being able to solve quadratic expressions, we can gain insights into these phenomena and make predictions about their behavior.

Tips and Tricks

Here are some tips and tricks to help you solve quadratic expressions:

  • Use the formula: (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2 to expand quadratic expressions.
  • Simplify the expression: by evaluating products and combining like terms.
  • Check your work: by plugging in values for the variable to ensure that the expression is true.

Common Mistakes

Here are some common mistakes to avoid when solving quadratic expressions:

  • Not using the formula: to expand quadratic expressions.
  • Not simplifying the expression: by evaluating products and combining like terms.
  • Not checking your work: by plugging in values for the variable to ensure that the expression is true.

Conclusion

Frequently Asked Questions

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial expression of degree two, which means it has a variable raised to the power of two. The general form of a quadratic expression is ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I expand a quadratic expression?

A: To expand a quadratic expression, you can use the formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2. For example, if you have the expression (x+6)2(x+6)^2, you can expand it as follows:

(x+6)2=x2+2(x)(6)+62(x+6)^2 = x^2 + 2(x)(6) + 6^2

Q: How do I simplify a quadratic expression?

A: To simplify a quadratic expression, you can evaluate the products and combine like terms. For example, if you have the expression x2+12x+36x^2 + 12x + 36, you can simplify it by combining the like terms:

x2+12x+36=x2+12x+36x^2 + 12x + 36 = x^2 + 12x + 36

Q: What is the difference between a quadratic expression and a linear expression?

A: A quadratic expression is a polynomial expression of degree two, while a linear expression is a polynomial expression of degree one. In other words, a quadratic expression has a variable raised to the power of two, while a linear expression has a variable raised to the power of one.

Q: Can I use the quadratic formula to solve quadratic expressions?

A: Yes, you can use the quadratic formula to solve quadratic expressions. The quadratic formula is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the quadratic formula used for?

A: The quadratic formula is used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It can be used to find the solutions to quadratic equations, which can be used to model real-world phenomena.

Q: Can I use the quadratic formula to solve quadratic expressions with complex solutions?

A: Yes, you can use the quadratic formula to solve quadratic expressions with complex solutions. The quadratic formula can be used to find the complex solutions to quadratic equations, which can be used to model real-world phenomena.

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

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

  • Not using the formula to expand quadratic expressions.
  • Not simplifying the expression by evaluating products and combining like terms.
  • Not checking your work by plugging in values for the variable to ensure that the expression is true.

Q: How can I practice solving quadratic expressions?

A: You can practice solving quadratic expressions by working through examples and exercises. You can also use online resources, such as math websites and apps, to practice solving quadratic expressions.

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

A: Quadratic expressions have many real-world applications, including:

  • Modeling the motion of objects.
  • Modeling the growth of populations.
  • Modeling the behavior of electrical circuits.
  • Modeling the behavior of financial systems.

Conclusion

In conclusion, quadratic expressions are a fundamental concept in mathematics and science. By understanding how to expand and simplify quadratic expressions, we can gain insights into real-world phenomena and make predictions about their behavior. By following the tips and tricks outlined in this article, we can avoid common mistakes and ensure that our solutions are accurate and reliable.