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How do you factor \[16{{x}^{2}}-36\] using the difference of squares?

Last updated date: 23rd Feb 2024
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IVSAT 2024
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Hint: The expressions which have one square term being subtracted from another square term is called difference of squares. The algebraic form of this is \[{{a}^{2}}-{{b}^{2}}\]. The factored form of these types of expression is \[\left( a+b \right)\left( a-b \right)\].

Complete step by step solution:
The given expression is \[16{{x}^{2}}-36\]. It has two terms; the first term is \[16{{x}^{2}}\] and the second term is 36.
As we know that 16 is square of 4, the first term can also be written as \[{{4}^{2}}{{x}^{2}}\]. Using the algebraic property, \[{{a}^{m}}{{b}^{m}}={{\left( ab \right)}^{m}}\]. This term can be written as \[{{\left( 4x \right)}^{2}}\]. The second term is 36, we know that 36 is a square of 6. This term can be written as \[{{6}^{2}}\]. Using this simplification in the given expression, it can be written as \[16{{x}^{2}}-36={{\left( 4x \right)}^{2}}-{{6}^{2}}\].
As we can see that this expression is evaluating the difference of two square terms, it is different from square form. We know that the difference of square expression \[{{a}^{2}}-{{b}^{2}}\] is factorized as \[\left( a+b \right)\left( a-b \right)\]. Here, we have a, and b are \[4x\] and 6 respectively. Substituting the values in the expansion, we get
\[\Rightarrow {{\left( 4x \right)}^{2}}-{{6}^{2}}=\left( 4x+6 \right)\left( 4x-6 \right)\]

Hence the factored form of the given expression is \[\left( 4x+6 \right)\left( 4x-6 \right)\].

Note: To solve these types of problems one should know the difference of square form, and its factored form. There are many other special expression forms like this such as, difference of cubes, addition of cubes. Algebraic form of difference of cubes and its expansion is \[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+{{b}^{2}}+ab \right)\]. Similarly, for the addition of cubes, it is \[{{a}^{3}}+{{b}^{3}}=\left( a+b \right)\left( {{a}^{2}}+{{b}^{2}}-ab \right)\].