Problem 186 Factor completely using the diff... [FREE SOLUTION] (2024)

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Chapter 6: Problem 186

Factor completely using the difference of squares pattern, if possible. $$ 36 p^{2}-49 q^{2} $$

Short Answer

Expert verified

(6p + 7q)(6p - 7q) .

Step by step solution


Identify terms

Observe the expression: [36 p^{2} - 49 q^{2} ]. Identify each term as a perfect square.


Express as squares

Write each term as a square: [36 p^{2} = (6p)^2 ] and [49 q^{2} = (7q)^2 ].


Apply difference of squares formula

Use the difference of squares formula: [a^2 - b^2 = (a + b)(a - b) ]. Here, [a = 6p ] and [b = 7q ].


Substitute and simplify

Substitute [a = 6p ] and [b = 7q ] into the formula to get: [(6p + 7q)(6p - 7q) ].

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares

The 'difference of squares' is a powerful tool in algebra for factorizing certain kinds of expressions. It applies to expressions that can be written in the form \[ a^2 - b^2 \].
The key to using this method is recognizing whether both terms in the expression are perfect squares.
Once identified, we can use the formula: \[ a^2 - b^2 = (a + b)(a - b) \].
In the given exercise, we have \[ 36p^2 - 49q^2 \]. Notice that both 36 and 49 are perfect squares, and so are the variables \[ p^2 \] and \[ q^2 \].
Rewriting, we find that \[ 36p^2 = (6p)^2 \] and \[ 49q^2 = (7q)^2 \].
Thus, \[ 36p^2 - 49q^2 \] can be transformed into \[ (6p)^2 - (7q)^2 \]. Using the difference of squares formula, we get \[ (6p + 7q)(6p - 7q) \].

Perfect Square

A perfect square is any number or expression that can be written as the square of another number or expression.
For instance, 36 is a perfect square because it can be written as \[ 6^2 \]. Similarly, \[ p^2 \] is a perfect square because it is \[ (p)^2 \].
Identifying perfect squares is crucial when dealing with algebraic expressions and factorization problems, particularly those involving the difference of squares.
When you encounter an expression such as \[ a^2 - b^2 \], check if \[ a^2 \] and \[ b^2 \] represent perfect squares.
In our exercise, \[ 36p^2 \] is a perfect square as it equals \[ (6p)^2 \], and similarly, \[ 49q^2 \] is a perfect square because it equals \[ (7q)^2 \].
Recognizing perfect squares can simplify complex algebraic expressions and aid in various factorization techniques.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. They form the foundation of algebra and are used to represent mathematical relationships and problems.
For example, in the expression \[ 36p^2 - 49q^2 \], we deal with terms involving both numbers (coefficients) and variables.
To simplify or factorize such expressions, it helps to understand the properties of numbers and algebraic rules.
In our exercise, the expression is \[ 36p^2 - 49q^2 \]. This can be factorized using the difference of squares method since both terms are perfect squares: \[ (6p)^2 - (7q)^2 \].
Applying the difference of squares formula yields \[ (6p + 7q)(6p - 7q) \].
Understanding how to manipulate and factorize algebraic expressions is an essential skill in solving mathematical problems efficiently.

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Problem 186 Factor completely using the diff... [FREE SOLUTION] (3)

Most popular questions from this chapter

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ y^{2}-18 y+45 $$In the following exercises, factor each trinomial of the form \(x^{2}+b x y+cy^{2} .\) $$ x^{2}-2 x y-80 y^{2} $$In the following exercises, factor by grouping. $$ 4 x^{2}-36 x-3 x+27 $$In the following exercises, factor using the 'ac' method. $$ 90 n^{3}+42 n^{2}-216 n $$In the following exercises, factor using the 'ac' method. $$ 2 n^{2}-27 n-45 $$
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Problem 186 Factor completely using the diff... [FREE SOLUTION] (2024)


How to find factor of 186? ›

The factors of 186 are 1, 2, 3, 6, 31, 62, 93, 186. Therefore, 186 has 8 factors.

What can you multiply to get 186? ›

The factors of 186 can be listed as 1, 2, 3, 6, 31, 62, 93 and 186.

What are the multiples of 186? ›

The first 10 multiples of 186 are 186, 372, 558, 744, 930, 1116, 1302, 1488, 1674 and 1860.

What are the prime numbers of 186? ›

If we look closely at the 8 different factors of 186 (1, 2, 3, 6, 31, 62, 93 and 186) there are three prime numbers hidden in those 8 factors : 2 , 3 and 31.

How to work out 186 divided by 6? ›

Divide 186 by 6 to get 31. Divide 186 by 6 to get 31.

What is the multiplication of 186? ›

The table of 186 can be written as 186 times 1 is 186, 186 times 2 are 372, 186 times 3 are 558, 186 times 4 are 744, 186 times 5 are 930 and so on.

Is 186 prime or composite? ›

The number 186 is divisible by 2 other than 1 and itself, so it meets the definition of a composite number.

What are the factors of 186 and 403? ›

The factors of 186 are 1, 2, 3, 6, 31, 62, 93, 186, and the factors of 403 are 1, 13, 31, 403. Learn to determine the HCF value of the given numbers using prime factorisation, long division and listing common factors by referring to this article.

How do you find the percentage of 186? ›

Next, calculate the percentage of 186: divide 186 by 1% value (1.86), and you get 100.00% - it's your percentage grade.

Is 186 divisible by 2 or 3? ›

186 is divisible by 2 and 3; therefore, it is divisible by 6.

Which number 186 is divisible from? ›

186 is divisible by 1, 2, 3, 6, 31, 62, 93, 186.

What are the roots of 186? ›

Expressed in a radical form: √25 = 5. Therefore, solving for the Square Root of 186, we find that the square root of 186 is 13.638.

What is a factor of 186? ›

The factors of 186 are:

1, 2, 3, 6, 31, 62, 93, 186.

What are the factors of 185? ›

The factors of 185 are 1, 5, 37 and 185.

How is 11 not a prime number? ›

Yes, 11 is a prime number. The number 11 is divisible only by 1 and the number itself. For a number to be classified as a prime number, it should have exactly two factors. Since 11 has exactly two factors, i.e. 1 and 11, it is a prime number.

What is the formula to calculate factor? ›

This expresses the number of factors formula as, (a + 1) × (b + 1), where a, and b are the exponents obtained after the prime factorization of the given number. For example, let us find the total number of factors of the number 12. The prime factorization of 12 = 2 × 2 × 3.

How do you find the factor of a number? ›

If we can express the given number as the product of two whole numbers, then the numbers being multiplied are factors of the product. Thus, to find all the factors of a number, find all the pairs of numbers that, when multiplied, give the given number as a product. As a result, the factors of 8 are 1, 2, 4, 8.

What is the formula to factor? ›

Factoring formulas are used to write an algebraic expression as the product of two or more expressions. Some important factoring formulas are given as, (a + b)2 = a2 + 2ab + b. (a - b)2 = a2 - 2ab + b.

How do you factor out numbers? ›

How to Factor Out Numbers
  1. Determine a common factor. A common factor is 2.
  2. Divide each term by the common factor and write the results of the division in parentheses, with the factor out in front.
  3. Determine whether you can factor out any other terms. ...
  4. Simplify the answer.
Mar 26, 2016

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