Question

How do you factor $6{{x}^{2}}-13x+6$?

Answer 1

We are given $6{{x}^{2}}-13x+6$, we are asked to find the factor of this, to do so, we will first understand the type of equation we have. Once we get that, we will find the greatest common factor from each term then in the remaining term we factor using the middle term.
We will use $a\times b$ in such a way that its sum or difference from the ‘b’ of the equation $a{{x}^{2}}+bx+c$. Once we split that, we will unite all the factors of the equation and we will get the required answer.

Complete answer:
We are given $6{{x}^{2}}-13x+6$ and we are asked to find its factor.
To find the factor of the equation given, we can see that as the highest power of the equation is 2, it is a polynomial of degree 2. So, it’s a quadratic equation.
Now, to find the factor, we will first find the possible greatest common factor of all those. Now, in 6, 13 and 6 we can see that there is no common term. So, we cannot take anything common from the equation.
Now, we will use the middle term splitting method.
In the middle term splitting method, if we consider the equation, $a{{x}^{2}}+bx+c$, we can produce ‘a’ by ‘c’ and then factor ‘ac’ in such a way that if the product is ‘ac’ while the sum or difference is made up to ‘b’.
Now, we will perform the middle term splitting on $6{{x}^{2}}-13x+6$.
So, here we have, a=6, b=-13 and c=6.
We will use these values to find two terms which help us in splitting the middle term.
Now, we can see that $-9\times -4=36$ and $-9+\left( -4 \right)=-13$.
So, we will use this to split the middle term.
Hence, we can write $6{{x}^{2}}-13x+6$ as $6{{x}^{2}}+\left( -9-4 \right)x+6$.
On opening the bracket, we get,
$6{{x}^{2}}-9x-4x+6$
When we take common term from the first two terms and from the last two terms, we get,
$3x\left( 2x-3 \right)-\left( 2x-3 \right)$
We can further write it as $\left( 3x-2 \right)\left( 2x-3 \right)$
Hence, we get $6{{x}^{2}}-13x+6=\left( 3x-2 \right)\left( 2x-3 \right)$
So, the factor of $6{{x}^{2}}-13x+6$ is $\left( 3x-2 \right)\left( 2x-3 \right)$.

Note: While finding the middle term using the factor $a\times c$, we need to keep in mind that when the sign of ‘a’ and ‘c’ are same then, ‘b’ is obtained by addition only, if the sign of ‘a’ and ‘c’ are different then ‘b’ can be obtained using only subtraction. So, as we have a=6 and c=6 having the same sign, ‘b’ is obtained as (-9)+(-4)=-13, by addition of -9 and -4.