Question

How do you factor \[49{{x}^{2}}-4{{y}^{2}}\]?

Answer 1

Hint: Factorisation or factoring is defined as the breaking of an entity into a product of their factors which when multiplied together give the original number. In the factorisation method, we reduce any algebraic or quadratic equation into its simpler form, where the equations are represented as the product of their factors instead of expanding the brackets. We can factorise the difference of two squares as a product of two binomials with alternating signs in the middle, positive and negative. We can represent it in the form \[{{x}^{2}}-{{y}^{2}}=(x+y)(x-y)\]. This formula is applicable only if x and y are perfect squares.

Complete step by step answer:
In the given binomial \[49{{x}^{2}}-4{{y}^{2}}\]. We know that 49 and 4 are perfect squares. So, we can rewrite the equation as
\[\Rightarrow {{7}^{2}}{{x}^{2}}-{{2}^{2}}{{y}^{2}}\]
We know that, from the properties of exponents \[{{a}^{m}}{{b}^{m}}={{(ab)}^{m}}\]. From this property
\[{{7}^{2}}{{x}^{2}}\] can be written as \[{{(7x)}^{2}}\] and \[{{2}^{2}}{{y}^{2}}\] can be written as \[{{(2y)}^{2}}\].
Now, we can write the above equation as
\[\Rightarrow {{(7x)}^{2}}-{{(2y)}^{2}}\]
Now consider the formula \[{{x}^{2}}-{{y}^{2}}=(x+y)(x-y)\].
Proof of \[{{x}^{2}}-{{y}^{2}}=(x+y)(x-y)\]:
Consider right-hand side of the equation
\[\Rightarrow \]\[(x+y)(x-y)\]
Now using distributive property, we can split it as
\[\begin{align}
  & \Rightarrow x\left( x-y \right)+y(x-y) \\
 & \Rightarrow {{x}^{2}}-xy+y(x-y) \\
 & \Rightarrow {{x}^{2}}-xy+yx-{{y}^{2}} \\
\end{align}\]
Since commutative property is obeyed by multiplication.
\[\Rightarrow xy=yx\]
Therefore, the equation becomes
\[\Rightarrow {{x}^{2}}-{{y}^{2}}\]
The above equation is the right hand side of the formula.
Hence proved.
So, we can write \[{{x}^{2}}-{{y}^{2}}=(x+y)(x-y)\], from this we can write \[{{(7x)}^{2}}-{{(2y)}^{2}}\] as
\[(7x+2y)(7x-2y)\].
\[\Rightarrow (7x+2y)(7x-2y)={{(7x)}^{2}}-{{(2y)}^{2}}\]
 \[\Rightarrow (7x+2y)(7x-2y)=49{{x}^{2}}-4{{y}^{2}}\]
\[\therefore (7x+2y)(7x-2y)=49{{x}^{2}}-4{{y}^{2}}\] is the required answer.

Note:
We need to have enough knowledge on the methods of factoring an expression completely. For some methods it is better to remember formulae like the one we solved now. It is better if we avoid calculation or simplification mistakes.