Factor The Following Expression:$\[ X^2 + X - 90 \\]$\[(x - [?])(x + \square)\\]

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that allows us to rewrite an expression as a product of simpler expressions. In this article, we will focus on factoring the quadratic expression $x^2 + x - 90$ into the form $(x - a)(x + b)$, where $a$ and $b$ are integers.

Understanding the Basics of Factoring

Before we dive into factoring the given expression, let's review the basics of factoring quadratic expressions. A quadratic expression in the form of $ax^2 + bx + c$ can be factored into the form $(x - a)(x + b)$, where $a$ and $b$ are integers. The process of factoring involves finding two numbers whose product is equal to the constant term $c$ and whose sum is equal to the coefficient of the linear term $b$.

Factoring the Expression $x^2 + x - 90$

To factor the expression $x^2 + x - 90$, we need to find two numbers whose product is equal to $-90$ and whose sum is equal to $1$. Let's start by listing the factors of $-90$:

• $-90 = -1 \times 90$
• $-90 = -2 \times 45$
• $-90 = -3 \times 30$
• $-90 = -5 \times 18$
• $-90 = -6 \times 15$
• $-90 = -9 \times 10$
• $-90 = -10 \times 9$
• $-90 = -12 \times 7.5$
• $-90 = -15 \times 6$
• $-90 = -18 \times 5$
• $-90 = -20 \times 4.5$
• $-90 = -25 \times 3.6$
• $-90 = -30 \times 3$
• $-90 = -35 \times 2.57$
• $-90 = -45 \times 2$
• $-90 = -50 \times 1.8$
• $-90 = -90 \times 1$

Now, let's find the pair of numbers whose sum is equal to $1$:

• $-1 + 90 = 89$ (not equal to $1$)
• $-2 + 45 = 43$ (not equal to $1$)
• $-3 + 30 = 27$ (not equal to $1$)
• $-5 + 18 = 13$ (not equal to $1$)
• $-6 + 15 = 9$ (not equal to $1$)
• $-9 + 10 = 1$ (equal to $1$)

Therefore, the pair of numbers whose product is equal to $-90$ and whose sum is equal to $1$ is $-9$ and $10$. We can now write the factored form of the expression as:

$$x^2 + x - 90 = (x - 9)(x + 10)$$

Conclusion

In this article, we have factored the quadratic expression $x^2 + x - 90$ into the form $(x - a)(x + b)$, where $a$ and $b$ are integers. We have also reviewed the basics of factoring quadratic expressions and provided a step-by-step guide on how to factor the given expression. By following these steps, you can factor any quadratic expression into its factored form.

Common Mistakes to Avoid

When factoring quadratic expressions, there are several common mistakes to avoid. Here are a few:

• Not checking the signs: When factoring, it's essential to check the signs of the numbers. If the product is negative, the signs of the numbers must be different.
• Not checking the sum: When factoring, it's essential to check the sum of the numbers. The sum must be equal to the coefficient of the linear term.
• Not using the correct method: There are several methods for factoring quadratic expressions, including the factoring method, the quadratic formula, and the completing the square method. Make sure to use the correct method for the given expression.

Practice Problems

Here are a few practice problems to help you practice factoring quadratic expressions:

• Factor the expression $x^2 + 5x - 6$.
• Factor the expression $x^2 - 7x - 18$.
• Factor the expression $x^2 + 2x - 15$.

Solutions

Here are the solutions to the practice problems:

• $x^2 + 5x - 6 = (x + 6)(x - 1)$
• $x^2 - 7x - 18 = (x - 9)(x + 2)$
• $x^2 + 2x - 15 = (x + 5)(x - 3)$

Conclusion

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that allows us to rewrite an expression as a product of simpler expressions. In our previous article, we provided a step-by-step guide on how to factor the quadratic expression $x^2 + x - 90$. In this article, we will answer some of the most frequently asked questions about factoring quadratic expressions.

Q: What is factoring?

A: Factoring is the process of rewriting an expression as a product of simpler expressions. In the case of quadratic expressions, factoring involves expressing the expression as a product of two binomials.

Q: What are the different methods of factoring quadratic expressions?

A: There are several methods of factoring quadratic expressions, including:

• Factoring by grouping: This method involves grouping the terms of the expression into two groups and factoring out the greatest common factor of each group.
• Factoring by using the quadratic formula: This method involves using the quadratic formula to find the roots of the expression and then factoring the expression as a product of two binomials.
• Factoring by completing the square: This method involves completing the square of the expression to find the roots and then factoring the expression as a product of two binomials.

Q: How do I know which method to use?

A: The choice of method depends on the specific expression and the desired outcome. Here are some general guidelines:

• Use factoring by grouping when: The expression can be grouped into two groups, and the greatest common factor of each group can be factored out.
• Use factoring by using the quadratic formula when: The expression cannot be factored by grouping, and the quadratic formula is needed to find the roots.
• Use factoring by completing the square when: The expression can be rewritten as a perfect square trinomial, and completing the square is the most efficient method.

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

A: Here are some common mistakes to avoid when factoring quadratic expressions:

• Not checking the signs: When factoring, it's essential to check the signs of the numbers. If the product is negative, the signs of the numbers must be different.
• Not checking the sum: When factoring, it's essential to check the sum of the numbers. The sum must be equal to the coefficient of the linear term.
• Not using the correct method: There are several methods for factoring quadratic expressions, including the factoring method, the quadratic formula, and the completing the square method. Make sure to use the correct method for the given expression.

Q: How can I practice factoring quadratic expressions?

A: Here are some ways to practice factoring quadratic expressions:

• Practice problems: Try factoring a variety of quadratic expressions, including those with different coefficients and constants.
• Online resources: Use online resources, such as math websites and apps, to practice factoring quadratic expressions.
• Workbooks and textbooks: Use workbooks and textbooks to practice factoring quadratic expressions.

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

A: Factoring quadratic expressions has many real-world applications, including:

• Science and engineering: Factoring quadratic expressions is used to model and solve problems in science and engineering, such as the motion of objects and the design of structures.
• Economics: Factoring quadratic expressions is used to model and solve problems in economics, such as the behavior of supply and demand curves.
• Computer science: Factoring quadratic expressions is used to model and solve problems in computer science, such as the behavior of algorithms and the design of data structures.

Conclusion

In conclusion, factoring quadratic expressions is a fundamental concept in algebra that allows us to rewrite an expression as a product of simpler expressions. By following the steps outlined in this article, you can factor any quadratic expression into its factored form. Remember to check the signs and the sum of the numbers, and use the correct method for the given expression. With practice, you will become proficient in factoring quadratic expressions and be able to solve a wide range of problems.