Question

Solve ${x^4} - 10{x^2} + 9 = 0$

Answer 1

Hint: Take the given expression, and make it simpler by replacing the expression and split the middle term, find the factors and make the required term the subject by moving all the terms on the opposite side. For simplification put ${x^2} = y$ and then solve quadratic in y.

Complete step-by-step answer:
Take the given expression: ${x^4} - 10{x^2} + 9 = 0$
The above expression can be re-written as –
${({x^2})^2} - 10{x^2} + 9 = 0$
Let us assume that: ${x^2} = y$ …. (A)
and place the above expression –
${y^2} - 10y + 9 = 0$
Split the middle term for the above expression, split the middle term in such a way that the product of the first term and the last term.
${y^2} - 1y - 9y + 9 = 0$
Make the pair of first two terms and the last two terms in the above expression.
$\underline {{y^2} - 1y} - \underline {9y + 9} = 0$
Find the common multiple common from the paired terms.
$y(y - 1) - 9(y - 1) = 0$
Find the common term, common from the above expression-
$(y - 1)(y - 9) = 0$
There are two cases –
$y - 1 = 0$
Make the required term the subject, move the constant term on the opposite side. When you move any term from one side to the opposite side, the sign of the terms also changes. Negative term becomes positive.
$ \Rightarrow y = 1$
Place the value from the equation (A)
$ \Rightarrow {x^2} = 1$
Take square-root on both the sides –
$ \Rightarrow \sqrt {{x^2}} = \sqrt 1 $
Square and square root cancel each other.
$ \Rightarrow x = \pm 1$
$y - 9 = 0$
Make the required term the subject –
$ \Rightarrow y = 9$
Place the value from the equation (A)
$ \Rightarrow {x^2} = 9$
Take square-root on both the sides –
$ \Rightarrow \sqrt {{x^2}} = \sqrt 9 $
Square and square root cancel each other.
$ \Rightarrow x = \pm 3$
Hence, the required solution is $x = \pm 1, \pm 3$
So, the correct answer is “$x = \pm 1, \pm 3$”.

Note: Be careful about the sign convention while moving terms from one side to the opposite side. Remember the square of any term is always positive but the square-root of any term can be positive or negative. Square can be defined as the product of the same numbers.