Question

How do you factor the expression ${{x}^{2}}-10xy+24{{y}^{2}}$ ?

Answer 1

Hint: We are given equation as ${{x}^{2}}-10xy+24{{y}^{2}}$ , we are asked to find the factor of that expression, in order to do so we will find first what type of equation we have, then we will see how many variable are there in the equation, then we will split middle term to get the equation to have 4 terms. Hence we will use a grouping method to find the factor of the same, in this method we make pairs of 2-2 and find common terms.

Complete step-by-step answer:
We are given that we have ${{x}^{2}}-10xy+24{{y}^{2}}$, we are asked to find the factor there.
We can see that our equation is comprised of 2-variable ‘x’ and ‘y’ and also we can see that each term is of order 2 as ${{x}^{2}},xy,{{y}^{2}}$ we have combine power as 2, hence it means we have quadratic equation in 2-variable.
To factor the equation, we will split the middle term then we use a method of grouping.
We have ${{x}^{2}}-10xy+24{{y}^{2}}$.
Here we have $a=1,b=-10\text{ and }c=24$ .
We can see $a\times c=24$ and $b=-10$ .
If we look closely, we can see that there are term -6 and -4 such that –
$\Rightarrow -6+\left( -4 \right)=0\text{ and }\Rightarrow -6\times \left( -4 \right)=24$ .
So, we use them to split our middle terms.
So, we can write –
${{x}^{2}}-10xy+24{{y}^{2}}={{x}^{2}}+\left( -6-4 \right)xy+24{{y}^{2}}$
By opening brackets, we get –
$={{x}^{2}}-6xy-4xy+24{{y}^{2}}$ .
Now, by the method of graph we make groups from the first 2 terms and last 2-terms.
So, $\Rightarrow {{x}^{2}}-6xy-4xy+24{{y}^{2}}$
Taking common terms out, we get –
$\Rightarrow x\left( x-6y \right)-4y\left( x-6y \right)$
As $x-6y$ is common, so taking in out, we get –
$\Rightarrow \left( x-4y \right)\left( x-6y \right)$ .
Hence ${{x}^{2}}-10xy+24{{y}^{2}}$ is a factor as $\left( x-4y \right)\left( x-6y \right)$ .

Note: Method of grouping is done on the equation involving for more in this we have a pair of 2-2 and then take the term common to them, using this we simplify our terms.
Also splitting of middle terms is a very delicate method, for example we have $2{{x}^{2}}-5x+3$ .
Here $a=2,b=5,c=3$ .
So, $a\times c=2\times 3=6$ .
So, we can see that we can obtain middle term 5 as $5=6-1$ as well as $5=2+3$ . so, we need to be careful while choosing what is best suitable for the given equation.