Question

How do you solve \[4x^{3} – 12x^{2} – 3x + 9 = 0\] ?

Answer 1

Hint:In this question, we need to solve the given expression \[4x^{3} – 12x^{2} – 3x + 9 = 0\] and need to find the roots of the expression. First, we can factorise the given equation by taking the common terms outside. In order to find the roots of the equation, we need to represent a given polynomial equation as a product of two or more terms . We can find the value of \[x\] by splitting each expression to \[0\] . Then we need to rewrite the roots in the form of an algebraic formula . Using this, we can solve the given equation.

Complete step by step answer:
Given, \[4x^{3} - 12x^{2} - 3x + 9 = 0\]. Let us consider the given expression as \[f(x)\].
\[f\left( x \right) = 4x^{3} - 12x^{2} - 3x + 9 = 0\]
Now we need to factorise the given expression by taking the common terms outside.The expression consists of four terms. We can take \[4x^{2}\] common from the first two terms and also \[- 3\] common from the next two terms.Given, \[4x^{3} - 12x^{2} - 3x + 9 = 0\]. By taking the common terms outside we get,
\[\Rightarrow \ 4x^{2}(x - 3) - 3\left( x - 3 \right) = 0\]
Again by taking \[(x - 3)\] common we get,
\[\left( x - 3 \right)\left( 4x^{2} - 3 \right) = 0\]

In the above expression ,the product of two simpler linear expressions
is equal to \[0\] . Thus we can find the value of \[x\] by splitting
each expression to \[0\]. That is
\[\left( x - 3 \right) = 0\] and \[\left( 4x^{2} - 3 \right) = 0\]
On simplifying the term \[(x - 3)\ = 0\] ,
We get,
\[x = 3\]
Also on simplifying the term \[\left( 4x^{2} - 3 \right) = 0\]
We can rewrite the term, \[\left( 4x^{2} - 1 \right)\] as \[\left( \left( 2x^{2} \right)\left( \sqrt{3} \right)^{2} \right)\] which is in the form of \[\left( a^{2} - b^{2} \right)\]

We know that
\[\left( a^{2} - b^{2} \right) = \left( a + b \right)\left( a - b \right)\]
Thus we get,
\[\left( 2x + \sqrt{3} \right)\left( 2x - \sqrt{3} \right) = 0\]
Here we can find the value of \[x\] also by splitting each expression to
\[0\].
\[\left( 2x + \sqrt{3} \right) = 0\ and\ \left( 2x - \sqrt{3} \right) = 0\]
On simplifying we can,
\[2x = - \sqrt{3}\] and \[\ 2x = \sqrt{3}\]
Thus we get,
\[x = - \dfrac{\sqrt{3}}{2},\dfrac{\sqrt{3}}{2}\]
\[\therefore \ x = \pm \dfrac{\sqrt{3}}{2}\]

Therefore the roots of the equation \[4x^{3} - 12x^{2} - 3x + 9 = 0\]
are \[3\] and \[\pm \dfrac{\sqrt{3}}{2}\].

Note:The concept used in this question to solve the given equation is solutions of quadratic equations by factorization. Factorization is nothing but writing a whole number into smaller numbers of the same kind. In factorization, we need to represent a given polynomial equation as a product of two or more terms .We can also check whether our answer is correct or not by using factor theorem. Factor theorem states that, let \[f(x)\] be a polynomial equation if \[f(a)\ = 0\] then \[(x – a)\] is the factor of the polynomial \[f(x)\].