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Sagot :
The question is given to factorize the expression:
[tex]36\mleft(3x-2\mright)^2\: -\: 25\mleft(2-x\mright)^2[/tex]Given that:
[tex]\begin{gathered} 36=6^2 \\ 15=5^2 \end{gathered}[/tex]Therefore, the expression becomes:
[tex]\Rightarrow6^2(3x-2)^2\: -\: 5^2(2-x)^2[/tex]Recall the rule of exponents:
[tex]m^x\cdot n^x=(m\cdot n)^x[/tex]Hence, the expression can be rewritten to be:
[tex]\Rightarrow\lbrack6(3x-2)\rbrack^2\: -\: \lbrack5(2-x)\rbrack^2[/tex]Expand the terms in the brackets:
[tex]\begin{gathered} 6(3x-2)=18x-12 \\ 5(2-x)=10-5x \end{gathered}[/tex]Hence, we have the expression to be:
[tex]\Rightarrow(18x-12)^2-(10-5x)^2[/tex]Recall the Difference of Two Squares Formula, defined as:
[tex]x^2-y^2=(x-y)(x+y)[/tex]Hence, we have the expression to be:
[tex](18x-12)^2-(10-5x)^2=\lbrack(18x-12)-(10-5x)\rbrack\cdot\lbrack(18x-12)+(10-5x)\rbrack[/tex]Simplifying, we have:
[tex]\Rightarrow(18x-12-10+5x)\cdot(18x-12+10-5x)=(23x-22)(13x-2)[/tex]ANSWER:
[tex]36\mleft(3x-2\mright)^{2}-25\mleft(2-x\mright)^{2}=(23x-22)(13x-2)[/tex]
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