Factorize $x^2 + X - 12$.What Is The Value Of B B B And C C C In The Trinomial? B = B = \qquad B = C = C = \qquad C =

Introduction

In algebra, factorizing a quadratic expression is an essential skill that helps us simplify complex equations and solve problems more efficiently. In this article, we will focus on factorizing the quadratic expression $x^2 + x - 12$ and determine the values of $b$ and $c$ in the trinomial.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The quadratic expression we are dealing with is $x^2 + x - 12$, where $a = 1$, $b = 1$, and $c = -12$.

Factorizing the Quadratic Expression

To factorize the quadratic expression $x^2 + x - 12$, we need to find two numbers whose product is $-12$ and whose sum is $1$. These numbers are $4$ and $-3$, because $4 \times (-3) = -12$ and $4 + (-3) = 1$. Therefore, we can write the quadratic expression as $(x + 4)(x - 3)$.

Determining the Values of $b$ and $c$

Now that we have factorized the quadratic expression, we can determine the values of $b$ and $c$ in the trinomial. The trinomial is $(x + 4)(x - 3)$, which can be expanded as $x^2 + x - 12$. Comparing this with the original quadratic expression, we can see that $b = 1$ and $c = -12$.

Conclusion

In conclusion, factorizing the quadratic expression $x^2 + x - 12$ involves finding two numbers whose product is $-12$ and whose sum is $1$. These numbers are $4$ and $-3$, and the quadratic expression can be written as $(x + 4)(x - 3)$. The values of $b$ and $c$ in the trinomial are $b = 1$ and $c = -12$.

Step-by-Step Solution

Here is a step-by-step solution to factorize the quadratic expression $x^2 + x - 12$:

1. Write the quadratic expression: $x^2 + x - 12$
2. Find two numbers whose product is $-12$ and whose sum is $1$: $4$ and $-3$
3. Write the quadratic expression as a product of two binomials: $(x + 4)(x - 3)$
4. Expand the product: $x^2 + x - 12$
5. Determine the values of $b$ and $c$: $b = 1$ and $c = -12$

Example Problems

Here are some example problems to practice factorizing quadratic expressions:

• Factorize the quadratic expression $x^2 - 5x + 6$
• Factorize the quadratic expression $x^2 + 7x + 12$
• Factorize the quadratic expression $x^2 - 2x - 15$

Tips and Tricks

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

• Use the factoring method: Factorize the quadratic expression by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
• Use the quadratic formula: Use the quadratic formula to find the roots of the quadratic equation.
• Check your work: Check your work by expanding the product and simplifying the expression.

Common Mistakes

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

• Not using the factoring method: Not using the factoring method can lead to incorrect factorization.
• Not checking your work: Not checking your work can lead to incorrect factorization.
• Not using the quadratic formula: Not using the quadratic formula can lead to incorrect factorization.

Conclusion

Introduction

In our previous article, we discussed how to factorize the quadratic expression $x^2 + x - 12$ and determine the values of $b$ and $c$ in the trinomial. In this article, we will provide a Q&A guide to help you understand the concept of factorizing quadratic expressions and address any questions or concerns you may have.

Q: What is factorizing a quadratic expression?

A: Factorizing a quadratic expression involves expressing it as a product of two binomials. This is done by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, follow these steps:

1. Write the quadratic expression: Write the quadratic expression in the form $ax^2 + bx + c$.
2. Find two numbers whose product is $c$ and whose sum is $b$: Find two numbers whose product is the constant term $c$ and whose sum is the coefficient of the linear term $b$.
3. Write the quadratic expression as a product of two binomials: Write the quadratic expression as a product of two binomials using the numbers you found in step 2.
4. Expand the product: Expand the product to simplify the expression.

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

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

• Not using the factoring method: Not using the factoring method can lead to incorrect factorization.
• Not checking your work: Not checking your work can lead to incorrect factorization.
• Not using the quadratic formula: Not using the quadratic formula can lead to incorrect factorization.

Q: How do I use the quadratic formula to factorize a quadratic expression?

A: The quadratic formula is a method for finding the roots of a quadratic equation. To use the quadratic formula to factorize a quadratic expression, follow these steps:

1. Write the quadratic equation: Write the quadratic equation in the form $ax^2 + bx + c = 0$.
2. Plug in the values: Plug in the values of $a$, $b$, and $c$ into the quadratic formula.
3. Simplify the expression: Simplify the expression to find the roots of the quadratic equation.

Q: What are some tips and tricks for factorizing quadratic expressions?

A: Some tips and tricks for factorizing quadratic expressions include:

• Use the factoring method: Factorize the quadratic expression by finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
• Use the quadratic formula: Use the quadratic formula to find the roots of the quadratic equation.
• Check your work: Check your work by expanding the product and simplifying the expression.

Q: How do I determine the values of $b$ and $c$ in a trinomial?

A: To determine the values of $b$ and $c$ in a trinomial, follow these steps:

1. Write the trinomial: Write the trinomial in the form $ax^2 + bx + c$.
2. Find the values of $a$, $b$, and $c$: Find the values of $a$, $b$, and $c$ by comparing the trinomial with the quadratic expression.
3. Determine the values of $b$ and $c$: Determine the values of $b$ and $c$ by comparing the trinomial with the quadratic expression.

Q: What are some example problems for factorizing quadratic expressions?

A: Some example problems for factorizing quadratic expressions include:

• Factorize the quadratic expression $x^2 - 5x + 6$
• Factorize the quadratic expression $x^2 + 7x + 12$
• Factorize the quadratic expression $x^2 - 2x - 15$

Conclusion

In conclusion, factorizing quadratic expressions involves expressing them as a product of two binomials. By following the step-by-step guide and using the tips and tricks provided, you can factorize quadratic expressions with ease. Remember to check your work and use the quadratic formula to find the roots of the quadratic equation.