Select The Correct Answer.What Is The Factored Form Of This Expression? $x^2 - 12x + 36$A. $(x-6)(x+6)$ B. $ ( X − 12 ) ( X − 3 ) (x-12)(x-3) ( X − 12 ) ( X − 3 ) [/tex] C. $(x+6)^2$ D. $(x-6)^2$

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. In this article, we will explore the factored form of a given quadratic expression and learn how to identify the correct answer among the options provided.

What is Factoring?

Factoring is the process of expressing a quadratic expression as a product of two or more binomial expressions. This involves finding the factors of the quadratic expression that, when multiplied together, result in the original expression. Factoring quadratic expressions is essential in solving equations, graphing functions, and simplifying complex expressions.

The Given Expression

The given quadratic expression is:

x2−12x+36x^2 - 12x + 36

This expression can be factored using various methods, including the factoring by grouping method, the perfect square trinomial method, and the quadratic formula method.

Factoring by Grouping Method

To factor the given expression using the factoring by grouping method, we need to group the terms in pairs and look for common factors. The expression can be rewritten as:

x2−12x+36=(x2−12x)+36x^2 - 12x + 36 = (x^2 - 12x) + 36

Now, we can factor out the common factor from the first two terms:

(x2−12x)=x(x−12)(x^2 - 12x) = x(x - 12)

So, the expression becomes:

x(x−12)+36x(x - 12) + 36

Next, we can factor out the common factor from the last two terms:

36=36(1)36 = 36(1)

Now, we can rewrite the expression as:

x(x−12)+36(1)x(x - 12) + 36(1)

This can be further simplified to:

(x−6)(x−6)+36(1)(x - 6)(x - 6) + 36(1)

However, this is not the correct factored form of the expression. We need to look for another method to factor the expression.

Perfect Square Trinomial Method

The given expression can also be factored using the perfect square trinomial method. A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. The general form of a perfect square trinomial is:

a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2

To factor the given expression using this method, we need to identify the values of a and b. In this case, a = x and b = 6.

So, the expression becomes:

(x−6)2(x - 6)^2

This is the correct factored form of the expression.

Conclusion

In this article, we have learned how to factor a quadratic expression using the factoring by grouping method and the perfect square trinomial method. We have also identified the correct factored form of the given expression as $(x - 6)^2$. This is an essential concept in algebra that helps us simplify complex expressions and solve equations.

Answer

The correct answer is:

  • D. $(x-6)^2$

This is the factored form of the given expression.

Discussion

  • What is the factored form of the expression x2−12x+36x^2 - 12x + 36?
  • How can we factor a quadratic expression using the factoring by grouping method?
  • What is the perfect square trinomial method, and how can we use it to factor a quadratic expression?
  • What is the correct factored form of the expression x2−12x+36x^2 - 12x + 36?

References

Introduction

In our previous article, we explored the factored form of a given quadratic expression and learned how to identify the correct answer among the options provided. In this article, we will continue to delve deeper into the world of factoring quadratic expressions and answer some of the most frequently asked questions.

Q&A

Q: What is the factored form of the expression x2+14x+49x^2 + 14x + 49?

A: The factored form of the expression x2+14x+49x^2 + 14x + 49 is (x+7)2(x + 7)^2.

Q: How can we factor a quadratic expression using the factoring by grouping method?

A: To factor a quadratic expression using the factoring by grouping method, we need to group the terms in pairs and look for common factors. We can then factor out the common factor from each pair of terms.

Q: What is the perfect square trinomial method, and how can we use it to factor a quadratic expression?

A: The perfect square trinomial method is a technique used to factor quadratic expressions that can be expressed as the square of a binomial. To use this method, we need to identify the values of a and b in the expression a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a - b)^2.

Q: How can we determine if a quadratic expression is a perfect square trinomial?

A: To determine if a quadratic expression is a perfect square trinomial, we need to check if it can be expressed in the form a2−2ab+b2a^2 - 2ab + b^2. If it can, then it is a perfect square trinomial.

Q: What is the difference between factoring by grouping and factoring by perfect square trinomial?

A: Factoring by grouping involves grouping the terms in pairs and looking for common factors, while factoring by perfect square trinomial involves expressing the quadratic expression as the square of a binomial.

Q: How can we use factoring to solve quadratic equations?

A: Factoring can be used to solve quadratic equations by expressing the equation in the form (x−a)(x−b)=0(x - a)(x - b) = 0. We can then set each factor equal to zero and solve for x.

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

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

  • Not checking if the expression is a perfect square trinomial
  • Not factoring out the greatest common factor (GCF)
  • Not using the correct method for factoring

Conclusion

In this article, we have answered some of the most frequently asked questions about factoring quadratic expressions. We have also explored the different methods used to factor quadratic expressions, including factoring by grouping and factoring by perfect square trinomial. By understanding these methods and avoiding common mistakes, we can become more confident and proficient in factoring quadratic expressions.

Discussion

  • What are some other methods used to factor quadratic expressions?
  • How can we use factoring to solve quadratic equations?
  • What are some common mistakes to avoid when factoring quadratic expressions?

References

Practice Problems

  1. Factor the expression x2+16x+64x^2 + 16x + 64.
  2. Factor the expression x2−14x+49x^2 - 14x + 49.
  3. Factor the expression x2+10x+25x^2 + 10x + 25.

Answers

  1. (x+8)2(x + 8)^2
  2. (x−7)2(x - 7)^2
  3. (x+5)2(x + 5)^2

Conclusion

In this article, we have explored the factored form of a given quadratic expression and learned how to identify the correct answer among the options provided. We have also answered some of the most frequently asked questions about factoring quadratic expressions and provided practice problems for readers to try. By understanding the different methods used to factor quadratic expressions, we can become more confident and proficient in factoring quadratic expressions.