Factor This Trinomial Completely: − 10 X 2 + 36 X + 16 -10x^2 + 36x + 16 − 10 X 2 + 36 X + 16 .A. − 10 X 2 + 36 X + 16 − 2 ( 5 X 2 − 18 X − 8 ) − 2 ( 5 X + 1 ) ( X − 8 ) \begin{array}{l} -10x^2 + 36x + 16 \\ -2(5x^2 - 18x - 8) \\ -2(5x + 1)(x - 8) \end{array} − 10 X 2 + 36 X + 16 − 2 ( 5 X 2 − 18 X − 8 ) − 2 ( 5 X + 1 ) ( X − 8 ) ​ B.$ \begin{array}{l} -10x^2 + 36x + 16 \ -2(5x^2 - 16x - 8) \ -2(5x + 4)(x -

Introduction

Factoring a trinomial is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomials. In this article, we will focus on factoring the trinomial $-10x^2 + 36x + 16$. We will explore the different methods of factoring and provide a step-by-step guide on how to factor this trinomial completely.

Understanding the Trinomial

Before we dive into the factoring process, let's take a closer look at the trinomial $-10x^2 + 36x + 16$. This trinomial has three terms: a quadratic term $-10x^2$, a linear term $36x$, and a constant term $16$. The coefficient of the quadratic term is negative, which means that the trinomial will have two real roots.

Method 1: Factoring by Grouping

One of the most common methods of factoring a trinomial is by grouping. This method involves grouping the first two terms and the last two terms, and then factoring out the common factors.

Step 1: Group the first two terms

The first two terms are $-10x^2$ and $36x$. We can group these terms by factoring out the common factor $-10x$.

-10x^2 + 36x = -10x(x - 3.6)

Step 2: Group the last two terms

The last two terms are $36x$ and $16$. We can group these terms by factoring out the common factor $4$.

36x + 16 = 4(9x + 4)

Step 3: Factor out the common factor

Now that we have grouped the terms, we can factor out the common factor $-2$.

-10x^2 + 36x + 16 = -2(5x^2 - 18x - 8)

Step 4: Factor the quadratic expression

The quadratic expression $5x^2 - 18x - 8$ can be factored as $(5x + 1)(x - 8)$.

-10x^2 + 36x + 16 = -2(5x + 1)(x - 8)

Method 2: Factoring by Using the Quadratic Formula

Another method of factoring a trinomial is by using the quadratic formula. This method involves using the quadratic formula to find the roots of the quadratic expression.

Step 1: Find the roots of the quadratic expression

The quadratic expression is $5x^2 - 18x - 8$. We can find the roots of this expression by using the quadratic formula.

x = (-b ± √(b^2 - 4ac)) / 2a

Step 2: Plug in the values

We can plug in the values $a = 5$, $b = -18$, and $c = -8$ into the quadratic formula.

x = (18 ± √((-18)^2 - 4(5)(-8))) / 2(5)

Step 3: Simplify the expression

We can simplify the expression by evaluating the square root.

x = (18 ± √(324 + 160)) / 10
x = (18 ± √484) / 10
x = (18 ± 22) / 10

Step 4: Find the two roots

We can find the two roots by plugging in the values $x = (18 + 22) / 10$ and $x = (18 - 22) / 10$.

x = 40 / 10
x = 4
x = -4 / 10
x = -2 / 5

Step 5: Factor the trinomial

We can factor the trinomial by using the two roots.

-10x^2 + 36x + 16 = -2(5x + 4)(x - 2)

Conclusion

In this article, we have explored two methods of factoring a trinomial: factoring by grouping and factoring by using the quadratic formula. We have also provided a step-by-step guide on how to factor the trinomial $-10x^2 + 36x + 16$ completely. By following these steps, you can factor any trinomial and understand the underlying mathematics behind it.

Discussion

What are some common mistakes that people make when factoring a trinomial? How can we avoid these mistakes and ensure that we factor the trinomial correctly?

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Additional Resources

• [1] Khan Academy: Factoring Quadratic Expressions
• [2] MIT OpenCourseWare: Algebra
• [3] Wolfram Alpha: Factoring Trinomials
  Q&A: Factoring Trinomials ==========================

Introduction

Factoring trinomials is a fundamental concept in algebra that can be challenging for many students. In this article, we will provide a Q&A section to help you understand the concept of factoring trinomials and address some common questions and concerns.

Q: What is a trinomial?

A trinomial is a polynomial expression that consists of three terms. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I factor a trinomial?

There are several methods to factor a trinomial, including factoring by grouping, factoring by using the quadratic formula, and factoring by using the difference of squares. The method you choose will depend on the specific trinomial you are working with.

Q: What is factoring by grouping?

Factoring by grouping involves grouping the first two terms and the last two terms, and then factoring out the common factors. This method is useful when the trinomial can be written in the form $(ax + b)(cx + d)$.

Q: What is factoring by using the quadratic formula?

Factoring by using the quadratic formula involves using the quadratic formula to find the roots of the quadratic expression. This method is useful when the trinomial can be written in the form $ax^2 + bx + c = 0$.

Q: What is factoring by using the difference of squares?

Factoring by using the difference of squares involves using the difference of squares formula to factor the trinomial. This method is useful when the trinomial can be written in the form $a^2 - b^2$.

Q: How do I know which method to use?

The method you choose will depend on the specific trinomial you are working with. You can try using different methods to see which one works best for you.

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

Some common mistakes to avoid when factoring trinomials include:

• Not factoring out the greatest common factor (GCF)
• Not using the correct method for the specific trinomial
• Not checking the factored form to make sure it is correct

Q: How can I practice factoring trinomials?

You can practice factoring trinomials by working through examples and exercises in your textbook or online resources. You can also try using online tools or software to help you practice factoring trinomials.

Q: What are some real-world applications of factoring trinomials?

Factoring trinomials has many real-world applications, including:

• Solving systems of equations
• Finding the roots of a quadratic equation
• Factoring polynomials in algebraic geometry

Conclusion

In this article, we have provided a Q&A section to help you understand the concept of factoring trinomials and address some common questions and concerns. We hope this article has been helpful in your understanding of factoring trinomials.

Discussion

What are some other questions you have about factoring trinomials? How can we help you better understand this concept?

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Additional Resources

• [1] Khan Academy: Factoring Quadratic Expressions
• [2] MIT OpenCourseWare: Algebra
• [3] Wolfram Alpha: Factoring Trinomials