EXERCISE 7(a) Factorize The Following:1. $x^2 - 1$2. $x^2 - 4$3. $a^2 - 16$4. $p^2 - 25$5. $m^2 - 36$6. $4x^2 - 1$7. $16x^2 - 9$8. $81y^2 - 16$9. $49n^2 - 121$10.

Introduction

Factorizing 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 explore the factorization of various quadratic expressions, including perfect square trinomials, difference of squares, and other types of quadratic expressions.

Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. The general form of a perfect square trinomial is:

$$a^2 + 2ab + b^2 = (a + b)^2$$

or

$$a^2 - 2ab + b^2 = (a - b)^2$$

We can use this formula to factorize the following quadratic expressions:

1. $x^2 - 1$

We can rewrite $x^2 - 1$ as $(x)^2 - (1)^2$. Using the formula for the difference of squares, we get:

$$x^2 - 1 = (x - 1)(x + 1)$$

2. $x^2 - 4$

We can rewrite $x^2 - 4$ as $(x)^2 - (2)^2$. Using the formula for the difference of squares, we get:

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

3. $a^2 - 16$

We can rewrite $a^2 - 16$ as $(a)^2 - (4)^2$. Using the formula for the difference of squares, we get:

$$a^2 - 16 = (a - 4)(a + 4)$$

4. $p^2 - 25$

We can rewrite $p^2 - 25$ as $(p)^2 - (5)^2$. Using the formula for the difference of squares, we get:

$$p^2 - 25 = (p - 5)(p + 5)$$

5. $m^2 - 36$

We can rewrite $m^2 - 36$ as $(m)^2 - (6)^2$. Using the formula for the difference of squares, we get:

$$m^2 - 36 = (m - 6)(m + 6)$$

Difference of Squares

A difference of squares is a quadratic expression that can be factored into the product of two binomials. The general form of a difference of squares is:

$$a^2 - b^2 = (a + b)(a - b)$$

We can use this formula to factorize the following quadratic expressions:

6. $4x^2 - 1$

We can rewrite $4x^2 - 1$ as $(2x)^2 - (1)^2$. Using the formula for the difference of squares, we get:

$$4x^2 - 1 = (2x - 1)(2x + 1)$$

7. $16x^2 - 9$

We can rewrite $16x^2 - 9$ as $(4x)^2 - (3)^2$. Using the formula for the difference of squares, we get:

$$16x^2 - 9 = (4x - 3)(4x + 3)$$

8. $81y^2 - 16$

We can rewrite $81y^2 - 16$ as $(9y)^2 - (4)^2$. Using the formula for the difference of squares, we get:

$$81y^2 - 16 = (9y - 4)(9y + 4)$$

9. $49n^2 - 121$

We can rewrite $49n^2 - 121$ as $(7n)^2 - (11)^2$. Using the formula for the difference of squares, we get:

$$49n^2 - 121 = (7n - 11)(7n + 11)$$

Other Types of Quadratic Expressions

In addition to perfect square trinomials and difference of squares, there are other types of quadratic expressions that can be factored using various techniques. For example:

10. $x^2 + 5x + 6$

We can factorize $x^2 + 5x + 6$ by finding two numbers whose product is $6$ and whose sum is $5$. These numbers are $2$ and $3$, so we can write:

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

11. $x^2 - 7x + 12$

We can factorize $x^2 - 7x + 12$ by finding two numbers whose product is $12$ and whose sum is $-7$. These numbers are $-3$ and $-4$, so we can write:

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

Conclusion

Q&A: Frequently Asked Questions about Factorizing Quadratic Expressions

Q: What is factorizing a quadratic expression?

A: Factorizing a quadratic expression involves expressing it as a product of two binomial expressions. This technique is essential in solving quadratic equations, simplifying expressions, and understanding the properties of quadratic functions.

Q: What are the different types of quadratic expressions that can be factored?

A: There are several types of quadratic expressions that can be factored, including:

• Perfect square trinomials
• Difference of squares
• Other types of quadratic expressions, such as those that can be factored using the method of grouping or the method of substitution.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you can use the formula:

$$a^2 + 2ab + b^2 = (a + b)^2$$

or

$$a^2 - 2ab + b^2 = (a - b)^2$$

Q: How do I factor a difference of squares?

A: To factor a difference of squares, you can use the formula:

$$a^2 - b^2 = (a + b)(a - b)$$

Q: What is the method of grouping?

A: The method of grouping involves factoring a quadratic expression by grouping the terms in pairs. This method is useful for factoring expressions that do not fit the perfect square trinomial or difference of squares formulas.

Q: What is the method of substitution?

A: The method of substitution involves factoring a quadratic expression by substituting a variable for one of the terms. This method is useful for factoring expressions that do not fit the perfect square trinomial or difference of squares formulas.

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

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

1. Group the terms in pairs.
2. Factor out the greatest common factor (GCF) from each pair.
3. Write the factored form of the expression.

Q: How do I factor a quadratic expression using the method of substitution?

A: To factor a quadratic expression using the method of substitution, you can follow these steps:

1. Substitute a variable for one of the terms.
2. Factor the resulting expression.
3. Substitute back in the original variable.

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 if the expression is a perfect square trinomial or a difference of squares before trying to factor it.
• Not factoring out the greatest common factor (GCF) from each pair of terms when using the method of grouping.
• Not substituting back in the original variable when using the method of substitution.

Q: How can I practice factoring quadratic expressions?

A: You can practice factoring quadratic expressions by:

• Working through example problems in a textbook or online resource.
• Creating your own practice problems and factoring them.
• Using online tools or software to generate practice problems and check your work.

Conclusion

Factorizing quadratic expressions is an essential skill in algebra that involves expressing a quadratic expression as a product of two binomial expressions. By mastering the techniques for factoring quadratic expressions, you can simplify complex expressions, solve quadratic equations, and gain a deeper understanding of the properties of quadratic functions.