Factor The Expression: $\[ X^2 - 10x + 24 \\]

Feb 26, 2025 by ADMIN 46 views

**Factoring Quadratic Expressions: A Step-by-Step Guide**

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomial expressions. This technique is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions. In this article, we will focus on factoring the given quadratic expression: $x^2 - 10x + 24$ . We will explore the different methods of factoring, including the use of the quadratic formula, and provide a step-by-step guide on how to factor the expression.

Understanding Quadratic Expressions

A quadratic expression is a polynomial expression of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. In the given expression, $x^2 - 10x + 24$ , the coefficients are $a = 1$ , $b = -10$ , and $c = 24$ .

Methods of Factoring Quadratic Expressions

There are several methods of factoring quadratic expressions, including:

Factoring by Grouping: This method involves grouping the terms of the quadratic expression into two pairs and then factoring out the greatest common factor (GCF) from each pair.
Factoring by Using the Quadratic Formula: This method involves using the quadratic formula to find the roots of the quadratic equation and then factoring the expression using the roots.
Factoring by Using the Difference of Squares: This method involves using the difference of squares formula to factor the quadratic expression.

Factoring the Given Expression

To factor the given expression, $x^2 - 10x + 24$ , we will use the method of factoring by grouping. This method involves grouping the terms of the quadratic expression into two pairs and then factoring out the greatest common factor (GCF) from each pair.

Step 1: Group the Terms

The given expression can be grouped as follows:

$x^2 - 10x + 24 = (x^2 - 8x) + (-2x + 24)$

Step 2: Factor Out the GCF

Now, we will factor out the greatest common factor (GCF) from each pair:

$(x^2 - 8x) + (-2x + 24) = x(x - 8) - 2(x - 8)$

Step 3: Factor Out the Common Binomial

Now, we will factor out the common binomial from each pair:

$x(x - 8) - 2(x - 8) = (x - 2)(x - 8)$

Therefore, the factored form of the given expression is $(x - 2)(x - 8)$ .

Conclusion

Factoring quadratic expressions is a crucial concept in algebra that involves expressing a quadratic expression as a product of two binomial expressions. In this article, we have focused on factoring the given quadratic expression, $x^2 - 10x + 24$ , using the method of factoring by grouping. We have also explored the different methods of factoring quadratic expressions, including the use of the quadratic formula and the difference of squares formula. By following the step-by-step guide provided in this article, readers can learn how to factor quadratic expressions and apply this technique to solve quadratic equations and simplify expressions.

Example Problems

Problem 1

Factor the quadratic expression: $x^2 + 5x + 6$

Solution

To factor the given expression, we will use the method of factoring by grouping. This method involves grouping the terms of the quadratic expression into two pairs and then factoring out the greatest common factor (GCF) from each pair.

$x^2 + 5x + 6 = (x^2 + 2x) + (3x + 6)$

Now, we will factor out the greatest common factor (GCF) from each pair:

$(x^2 + 2x) + (3x + 6) = x(x + 2) + 3(x + 2)$

Now, we will factor out the common binomial from each pair:

$x(x + 2) + 3(x + 2) = (x + 3)(x + 2)$

Therefore, the factored form of the given expression is $(x + 3)(x + 2)$ .

Problem 2

Factor the quadratic expression: $x^2 - 7x + 12$

Solution

$x^2 - 7x + 12 = (x^2 - 4x) + (-3x + 12)$

Now, we will factor out the greatest common factor (GCF) from each pair:

$(x^2 - 4x) + (-3x + 12) = x(x - 4) - 3(x - 4)$

Now, we will factor out the common binomial from each pair:

$x(x - 4) - 3(x - 4) = (x - 3)(x - 4)$

Therefore, the factored form of the given expression is $(x - 3)(x - 4)$ .

Tips and Tricks

Use the Quadratic Formula: The quadratic formula can be used to find the roots of a quadratic equation and then factor the expression using the roots.
Use the Difference of Squares: The difference of squares formula can be used to factor quadratic expressions that can be written in the form of a difference of squares.
Check for Common Factors: Before factoring a quadratic expression, check for common factors and factor them out first.

Conclusion

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two binomial expressions. In our previous article, we provided a step-by-step guide on how to factor quadratic expressions using the method of factoring by grouping. In this article, we will provide a Q&A guide to help readers understand the concept of factoring quadratic expressions and apply it to solve quadratic equations and simplify expressions.

Q&A

Q: What is factoring a quadratic expression?

A: Factoring a quadratic expression involves expressing it as a product of two binomial expressions.

Q: Why is factoring a quadratic expression important?

A: Factoring a quadratic expression is important because it helps us to solve quadratic equations and simplify expressions.

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 quadratic expression into two pairs and then factoring out the greatest common factor (GCF) from each pair.
Factoring by Using the Quadratic Formula: This method involves using the quadratic formula to find the roots of the quadratic equation and then factoring the expression using the roots.
Factoring by Using the Difference of Squares: This method involves using the difference of squares formula to factor the quadratic expression.

Q: How do I factor a quadratic expression using the method of factoring by grouping?

A: To factor a quadratic expression using the method of factoring by grouping, follow these steps:

Group the terms of the quadratic expression into two pairs.
Factor out the greatest common factor (GCF) from each pair.
Factor out the common binomial from each pair.

Q: How do I factor a quadratic expression using the quadratic formula?

A: To factor a quadratic expression using the quadratic formula, follow these steps:

Use the quadratic formula to find the roots of the quadratic equation.
Factor the expression using the roots.

Q: How do I factor a quadratic expression using the difference of squares?

A: To factor a quadratic expression using the difference of squares, follow these steps:

Write the quadratic expression in the form of a difference of squares.
Factor the expression using the difference of squares formula.

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 for common factors: Before factoring a quadratic expression, check for common factors and factor them out first.
Not using the correct method: Use the correct method of factoring for the given quadratic expression.
Not simplifying the expression: Simplify the expression after factoring to ensure that it is in its simplest form.

Conclusion

Factoring quadratic expressions is a crucial concept in algebra that involves expressing a quadratic expression as a product of two binomial expressions. In this article, we have provided a Q&A guide to help readers understand the concept of factoring quadratic expressions and apply it to solve quadratic equations and simplify expressions. By following the steps and tips provided in this article, readers can learn how to factor quadratic expressions and apply this technique to solve quadratic equations and simplify expressions.

Example Problems

Problem 1

Factor the quadratic expression: $x^2 + 5x + 6$

Solution

$x^2 + 5x + 6 = (x^2 + 2x) + (3x + 6)$

Now, we will factor out the greatest common factor (GCF) from each pair:

$(x^2 + 2x) + (3x + 6) = x(x + 2) + 3(x + 2)$

Now, we will factor out the common binomial from each pair:

$x(x + 2) + 3(x + 2) = (x + 3)(x + 2)$

Therefore, the factored form of the given expression is $(x + 3)(x + 2)$ .

Problem 2

Factor the quadratic expression: $x^2 - 7x + 12$

Solution

$x^2 - 7x + 12 = (x^2 - 4x) + (-3x + 12)$

Now, we will factor out the greatest common factor (GCF) from each pair:

$(x^2 - 4x) + (-3x + 12) = x(x - 4) - 3(x - 4)$

Now, we will factor out the common binomial from each pair:

$x(x - 4) - 3(x - 4) = (x - 3)(x - 4)$

Therefore, the factored form of the given expression is $(x - 3)(x - 4)$ .

Tips and Tricks

Use the Quadratic Formula: The quadratic formula can be used to find the roots of a quadratic equation and then factor the expression using the roots.
Use the Difference of Squares: The difference of squares formula can be used to factor quadratic expressions that can be written in the form of a difference of squares.
Check for Common Factors: Before factoring a quadratic expression, check for common factors and factor them out first.

Conclusion

Factoring quadratic expressions is a crucial concept in algebra that involves expressing a quadratic expression as a product of two binomial expressions. In this article, we have provided a Q&A guide to help readers understand the concept of factoring quadratic expressions and apply it to solve quadratic equations and simplify expressions. By following the steps and tips provided in this article, readers can learn how to factor quadratic expressions and apply this technique to solve quadratic equations and simplify expressions.