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

Last updated date: 16th Jul 2024
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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.

Given equation is ${x^4} - 18{x^2} + 81$
${x^4} - 18{x^2} + 81 \\ \Rightarrow {x^4} - (9 + 9){x^2} + 81 \\ \Rightarrow {x^4} - 9{x^2} - 9{x^2} + 81 \\ \Rightarrow {x^2}({x^2} - 9) - 9({x^2} - 9) \\ \Rightarrow ({x^2} - 9)({x^2} - 9) \\ \Rightarrow ({x^2} - {3^2})({x^2} - {3^2}) \\ \therefore (x - 3)(x + 3)(x - 3)(x + 3) \\$
Hence, the factors are $(x - 3),(x + 3),(x - 3),(x + 3)$.