Fully Factorize $12x - 28 + X^2$.

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Introduction

In algebra, factorization is a process of expressing an algebraic expression as a product of simpler expressions. It is a crucial concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on fully factorizing the given quadratic expression $12x - 28 + x^2$. We will use various techniques, including grouping, factoring out common factors, and using the quadratic formula.

Understanding the Expression

Before we proceed with factorization, let's analyze the given expression. The expression is a quadratic expression in the form of $ax^2 + bx + c$. In this case, the coefficients are $a = 1$, $b = 12$, and $c = -28$. The expression can be written as:

$$12x - 28 + x^2$$

Grouping and Factoring Out Common Factors

One of the techniques used in factorization is grouping. Grouping involves dividing the expression into two or more groups and then factoring out common factors from each group. In this case, we can group the terms as follows:

$$12x - 28 + x^2 = (12x + x^2) - 28$$

Now, we can factor out the common factor $x$ from the first group:

$$(12x + x^2) - 28 = x(12 + x) - 28$$

Factoring Out Common Factors

Another technique used in factorization is factoring out common factors. In this case, we can factor out the common factor $4$ from the first group:

$$x(12 + x) - 28 = 4x(3 + x) - 28$$

Using the Quadratic Formula

The quadratic formula is a powerful tool used in factorization. It states that if $ax^2 + bx + c = 0$, then the solutions are given by:

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

In this case, we can rewrite the expression as a quadratic equation:

$$x^2 + 12x - 28 = 0$$

Now, we can use the quadratic formula to find the solutions:

$$x = \frac{-12 \pm \sqrt{12^2 - 4(1)(-28)}}{2(1)}$$

Simplifying the Expression

After using the quadratic formula, we get:

$$x = \frac{-12 \pm \sqrt{144 + 112}}{2}$$

$$x = \frac{-12 \pm \sqrt{256}}{2}$$

$$x = \frac{-12 \pm 16}{2}$$

Finding the Solutions

Now, we can find the solutions by simplifying the expression:

$$x = \frac{-12 + 16}{2}$$

$$x = \frac{4}{2}$$

$$x = 2$$

$$x = \frac{-12 - 16}{2}$$

$$x = \frac{-28}{2}$$

$$x = -14$$

Conclusion

In this article, we fully factorized the given quadratic expression $12x - 28 + x^2$. We used various techniques, including grouping, factoring out common factors, and using the quadratic formula. We found the solutions to the expression and simplified the expression to its final form. Factorization is a crucial concept in mathematics, and it has numerous applications in various fields. We hope that this article has provided a clear understanding of the concept of factorization and its applications.

Final Answer

The final answer to the problem is:

x^2 + 12x - 28 = (x + 14)(x - 2)$<br/> # Fully Factorize $12x - 28 + x^2$: Q&A ## Introduction In our previous article, we fully factorized the given quadratic expression $12x - 28 + x^2$. We used various techniques, including grouping, factoring out common factors, and using the quadratic formula. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic. ## Q: What is factorization? A: Factorization is a process of expressing an algebraic expression as a product of simpler expressions. It is a crucial concept in mathematics, and it has numerous applications in various fields. ## Q: Why is factorization important? A: Factorization is important because it helps us to simplify complex expressions and solve equations. It also helps us to identify the roots of a quadratic equation, which is essential in many real-world applications. ## Q: What are the different techniques used in factorization? A: There are several techniques used in factorization, including: * Grouping: This involves dividing the expression into two or more groups and then factoring out common factors from each group. * Factoring out common factors: This involves factoring out common factors from the expression. * Using the quadratic formula: This involves using the quadratic formula to find the solutions to a quadratic equation. ## Q: How do I factorize a quadratic expression? A: To factorize a quadratic expression, you can follow these steps: 1. Identify the coefficients of the quadratic expression. 2. Use the quadratic formula to find the solutions. 3. Simplify the expression to its final form. ## Q: What is the quadratic formula? A: The quadratic formula is a powerful tool used in factorization. It states that if $ax^2 + bx + c = 0$, then the solutions are given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the coefficients of the quadratic expression and plug them into the formula. Then, simplify the expression to find the solutions.

Q: What are the solutions to the expression $12x - 28 + x^2$?

A: The solutions to the expression $12x - 28 + x^2$ are $x = 2$ and $x = -14$.

Q: How do I simplify the expression $12x - 28 + x^2$?

A: To simplify the expression $12x - 28 + x^2$, you can use the quadratic formula to find the solutions. Then, simplify the expression to its final form.

Q: What is the final form of the expression $12x - 28 + x^2$?

A: The final form of the expression $12x - 28 + x^2$ is $(x + 14)(x - 2)$.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts and provide additional information on the topic of fully factorizing the given quadratic expression $12x - 28 + x^2$. We hope that this article has provided a clear understanding of the concept of factorization and its applications.

Final Answer

The final answer to the problem is:

$$x^2 + 12x - 28 = (x + 14)(x - 2)$$