Answer :
To factorise the given expressions, we are dealing with a special form known as the difference of squares. The formula for the difference of squares is:
[tex]a^2 - b^2 = (a + b)(a - b)[/tex]
We will apply this formula to each of the given expressions:
Expression: [tex]36(x + y)^2 - 4[/tex]
First, rewrite the expression in the form of [tex]a^2 - b^2[/tex]:
- [tex]a^2 = 36(x + y)^2 \rightarrow a = 6(x + y)[/tex]
- [tex]b^2 = 4 \rightarrow b = 2[/tex]
Using the difference of squares formula:
[tex]36(x + y)^2 - 4 = (6(x + y) + 2)(6(x + y) - 2)[/tex]
Expression: [tex]4(m + n)^2 - 49[/tex]
Rewrite the expression:
- [tex]a^2 = 4(m + n)^2 \rightarrow a = 2(m + n)[/tex]
- [tex]b^2 = 49 \rightarrow b = 7[/tex]
Apply the formula:
[tex]4(m + n)^2 - 49 = (2(m + n) + 7)(2(m + n) - 7)[/tex]
Expression: [tex]16(d + e)^2 - 81[/tex]
Rewrite the expression:
- [tex]a^2 = 16(d + e)^2 \rightarrow a = 4(d + e)[/tex]
- [tex]b^2 = 81 \rightarrow b = 9[/tex]
Apply the formula:
[tex]16(d + e)^2 - 81 = (4(d + e) + 9)(4(d + e) - 9)[/tex]
Expression: [tex]25(o + p)^2 - 81[/tex]
Rewrite the expression:
- [tex]a^2 = 25(o + p)^2 \rightarrow a = 5(o + p)[/tex]
- [tex]b^2 = 81 \rightarrow b = 9[/tex]
Apply the formula:
[tex]25(o + p)^2 - 81 = (5(o + p) + 9)(5(o + p) - 9)[/tex]
Expression: [tex]49(v + \omega)^2 - 16[/tex]
Rewrite the expression:
- [tex]a^2 = 49(v + \omega)^2 \rightarrow a = 7(v + \omega)[/tex]
- [tex]b^2 = 16 \rightarrow b = 4[/tex]
Apply the formula:
[tex]49(v + \omega)^2 - 16 = (7(v + \omega) + 4)(7(v + \omega) - 4)[/tex]
Expression: [tex](q + r)^2 - 16[/tex]
Rewrite the expression:
- [tex]a^2 = (q + r)^2 \rightarrow a = q + r[/tex]
- [tex]b^2 = 16 \rightarrow b = 4[/tex]
Apply the formula:
[tex](q + r)^2 - 16 = ((q + r) + 4)((q + r) - 4)[/tex]
These steps will help you understand how to factor expressions using the difference of squares.
