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Find the factors of the given quadratic equation and solve it \[{x^4} - 18{x^2} + 81\] ?

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Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint:Number of factors for any algebraic equation depends upon the degree of the equation, degree of equation means the maximum power an equation has either the equation have one variable or having multiple variables. For example: For any equation \[a{x^2} + bx + c\] the degree of equation is two hence it has two factors.

Formulae Used:
Mid term splitting rule is used here,
For any equation \[{a^2} + ab + c\]
We have to split “b” in say “d” and “e” such that \[d \times e = ac\] and \[d + e = b\] for the above sample equation.Now accordingly take the common factors about in a bracket and rest in another bracket and your answer will come i.e. factors will come from the taken equation.

Complete step by step answer:
Given equation is \[{x^4} - 18{x^2} + 81\]
Using mid term splitting rule, we have to break the middle term, we get:
 \[{x^4} - 18{x^2} + 81 \\
\Rightarrow {x^4} - (9 + 9){x^2} + 81 \\
\Rightarrow {x^4} - 9{x^2} - 9{x^2} + 81 \\
\Rightarrow {x^2}({x^2} - 9) - 9({x^2} - 9) \\
\Rightarrow ({x^2} - 9)({x^2} - 9) \\
\Rightarrow ({x^2} - {3^2})({x^2} - {3^2}) \\
\therefore (x - 3)(x + 3)(x - 3)(x + 3) \\ \]
Hence, the factors are \[(x - 3),(x + 3),(x - 3),(x + 3)\].

Additional Information: You have to be careful while breaking the middle term and in calculating the product of the coefficient of first term and the last term, their respective signs have to be considered and accordingly signs between the split terms should be used.

Note:Midterm splitting is very easy technique to factories the equation, but in case it is not possible then you can adopt the maximum common factor technique in which you have to take common the maximum possible common factor from the equation and rest you can solve according to the same method used in the midterm common factor technique.
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