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Give the factors and solution of the expression \[3{x^2} - 27 = 0\]by any mathematical method?

Last updated date: 20th Jun 2024
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Hint:Here we have to factorize the question by using mathematical properties, such a quadratic equation can also be solved by the sridharacharya rule, and here you can use them for finding the solution of the equation. For finding factors you have to rearrange the term so as to obtain the factors by taking common or doing some adjustments.

Complete step by step answer:
Here, for the given equation we have to do some rearrangement by taking common of the number which is common in both terms, and then we can easily see that it is becoming an equation which can be easily solved by algebraic identity say:
\[ \Rightarrow {a^2} - {b^2} = (a + b)(a - b)\]
On solving we get:
\[3{x^2} - 27 = 0 \\
\Rightarrow 3({x^2} - 9) = 0 \\
\Rightarrow 3({x^2} - {3^2}) = 0 \\
\Rightarrow 3(x - 3)(x + 3) = 0 \\
\Rightarrow (x - 3)(x + 3) = 0 \\
\therefore x = 3, - 3 \]
So the solution for the above equation is \[3, - 3\].

Hence, the factors of the equation are \[(x + 3),(x - 3)\].

Additional Information:
Mid term split method is easy to find, but here it cant be applied because in the mid term equation the middle term is needed to break it into two parts to obtain the conditions but here for this question the middle term is not present.

Note: In two variable questions it will work but make the question very complicated, so you have to be careful while using the highest common factor method. This method is fast and easy to use but is applicable to certain specific question only and in this question it can be applied very easily.Example of this method could be a three variable algebraic equation like: \[xyz + xy \]