Question

How do you factor \[3{{x}^{2}}-2x\]?

Answer 1

Hint: We can only solve this question if we have better knowledge in the chapter of polynomials. In this question, for finding the factors of \[3{{x}^{2}}-2x\] , we will find the factor of \[3{{x}^{2}}\] and 2x. After that, we will look for the common factor of the equation. We will also see the alternative method of this question.

Complete step by step answer:
Let us solve this question.
As we know that, if x is multiplied two times, then we get the square of x.
Then, according to the polynomials
\[x\times x={{x}^{2}}\]
Because x is multiplied two times that’s why power of x is 2.
Then, we can factor \[3{{x}^{2}}\] as \[3\times x\times x\].
Therefore, we can write the equation \[3{{x}^{2}}-2x\] as \[3\times x\times x-2\times x\].
That is,
\[\Rightarrow 3{{x}^{2}}-2x=3\times x\times x-2\times x\]
Now, we can see in the above equation that there is a common factor of x.
Therefore, we can find out the factor x from the above equation.
Let us take out that factor.
\[3{{x}^{2}}-2x\] can also be written as \[x(3\times x-2)\]
\[\Rightarrow 3{{x}^{2}}-2x=x(3x-2)\]

Hence, the factors of \[3{{x}^{2}}-2x\] are \[x\] and \[3x-2\].

Note: We can solve this question by a different method.
Method 2 for solving this question.
We know that the simple formula for finding the roots of equation \[a{{x}^{2}}+bx+c=0\] is
 \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
And, one more thing we have to know that \[a{{x}^{2}}+bx+c=0\] also can be written as
\[\left\{ x-\left( \dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a} \right) \right\}\left\{ x-\left( \dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a} \right) \right\}\]
Therefore, in the equation \[3{{x}^{2}}-2x=0\] relating to the equation \[a{{x}^{2}}+bx+c=0\],
a=3, b=-2, and c=0
 Hence, for the equation \[3{{x}^{2}}-2x\] , roots will be
\[x=\dfrac{-(-2)\pm \sqrt{{{(-2)}^{2}}-4\times 3\times 0}}{2\times 3}\]
The above equation also can be written as
\[\Rightarrow x=\dfrac{2\pm \sqrt{4-0}}{6}\]
\[\Rightarrow x=\dfrac{2\pm 2}{6}\]
Hence, the values of x are \[\dfrac{2}{3}\] and \[0\].
Therefore, we can write the equation \[3{{x}^{2}}-2x=0\] as \[\left\{ x-\dfrac{2}{3} \right\}\left\{ x-0 \right\}=0\]
Which is also can be written as
\[\Rightarrow \left\{ 3x-2 \right\}\left\{ x-0 \right\}=0\]
\[\Rightarrow \left( 3x-2 \right)\left( x \right)=0\]
Hence, from here also, we get that the factor of \[3{{x}^{2}}-2x\] is \[3x-2\] and \[x\].
So, we can use this method too for solving this type of question.