Answers:
4a.) [tex]49x^2-36y^2[/tex]
4b.) [tex]9a^2-25b^2[/tex]
4c.) [tex]\frac{1}{25}x^2-49[/tex]
5b.) [tex]4y^2-4xy-5x^2[/tex]
Solution Steps:
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4a.) [tex]\bold{(7x-6y)(7x+6y)}[/tex]:
- Multiplication can be transformed into difference of squares using the rule: [tex](a-b)(a+b)=a^2-b^2[/tex].
- So change the equation using the rule:
- Expand [tex](7x)^2[/tex] and [tex](6y)^2[/tex]:
- Calculate [tex]7^2[/tex] and [tex]6^2[/tex]:
So the end equation would be: [tex]49x^2-36y^2[/tex].
4b.) [tex]\bold{(3a+5b)(3a-5b)}[/tex]:
- Multiplication can be transformed into difference of squares using the rule: [tex](a-b)(a+b)=a^2-b^2[/tex].
- So change the equation using the rule:
- Expand [tex](3a)^2[/tex] and [tex](5b)^2[/tex] :
- Calculate [tex]3^2[/tex] and [tex]5^2[/tex]:
So the end equation would be: [tex]9a^2-25b^2[/tex].
4c.) [tex]\bold{(\frac{1}{5}x-7)}[/tex] × [tex]\bold{(\frac{1}{5}+7)}[/tex]:
- Multiplication can be transformed into difference of squares using the rule: [tex](a-b)(a+b)=a^2-b^2[/tex].
- So change the equation using the rule:
- [tex](\frac{1}{5}x)^2-7^2[/tex]
- Expand [tex](\frac{1}{5}x)^2[/tex]:
- [tex](\frac{1}{5})^2x^2[/tex]
- Calculate [tex](\frac{1}{5})^2[/tex] and [tex]7^2[/tex] :
- [tex](\frac{1}{5})^2=\frac{1}{25}[/tex]
So the end equation would be: [tex]\frac{1}{25}x^2-49[/tex].
5b.) [tex]\bold{(4x-y)(4x+y)+(2x-y)^2-(5x+2y)(5x-2y)}[/tex]:
- Consider [tex](4x-y)(4x+y)[/tex]. Multiplication can be transformed into difference of squares using the rule: [tex](a-b)(a+b)=a^2-b^2[/tex].
- So change the equation using the rule:
- Expand [tex](4x)^2[/tex]:
- Calculate [tex]4^2[/tex]:
- Use binomial theorem [tex](a-b)^2=a^2-2ab+b^2[/tex] to expand [tex](2x-y)^2[/tex]:
- Combine [tex]-y^2[/tex] and [tex]y^2[/tex]:
- Consider [tex](5x+2y)(5x-2y)[/tex]. Multiplication can be transformed into difference of squares using the rule: [tex](a-b)(a+b)=a^2-b^2[/tex].
So change the equation using the rule:
- Expand [tex](5x)^2[/tex] and [tex](2y)^2[/tex]:
- Calculate [tex]5^2[/tex] and [tex]2^2[/tex]:
- Combine [tex]20x^2[/tex] and [tex]-25x^2[/tex]:
- [tex]20x^2+(-25x^2)=-5x^2[/tex]
So the end equation would be: [tex]4y^2-4xy-5x^2[/tex].
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