Question

How do you factor \[25{{x}^{2}}-40xy+16{{y}^{2}}\]?

Answer 1

Hint: In this problem, we have to find the factor of the given expression with two variables by factorization method. We can first split the middle term. We can factorize by expanding the middle term i.e. the x term with its coefficient in such a way that their addition is equal to the middle term i.e. -40xy, and multiplication is equal to \[25\times 16=400=-20\times -20\], which is the multiplication of first term and the last term. We can then take common terms outside to get the factors.

Complete step by step solution:
We know that the given expression is,
\[25{{x}^{2}}-40xy+16{{y}^{2}}\]…….. (1)
We can first split the middle term to form factors.
We have to expand the middle term i.e. the x term with its coefficient in such a way that their addition is equal to the middle term i.e. -40xy, and multiplication is equal to\[25\times 16=400=-20\times -20\]
\[\begin{align}
  & \Rightarrow 25\times 16=400=-20\times -20 \\
 & \Rightarrow -20-20=-40 \\
\end{align}\]
We can apply the above step in the equation (1), we get
\[25{{x}^{2}}-20xy-20xy+16{{y}^{2}}\]
We can now take the first two terms and the last two terms to take common terms outside, we get
\[\Rightarrow \left( 25{{x}^{2}}-20xy \right)+\left( -20xy+16{{y}^{2}} \right)\]
Now we can take the common terms outside, we get
\[\Rightarrow 5x\left( 5x-4y \right)-4y\left( 5x-4y \right)\]
We can again take the common factor first then the remaining terms to make a factor, we get
\[\Rightarrow \left( 5x-4y \right)\left( 5x-4y \right)\]

Therefore, the factors are \[{{\left( 5x-4y \right)}^{2}}\].

Note: We can see that, we are given an equation with two variables for which we have to find the factor. Students make mistakes while splitting the middle term such as its addition should be equal to the middle term itself and its multiplication should be equal to the multiplication of first and the last term.