Question

How do you factor $ {a^2} - 2ab + {b^2} $ ?

Answer 1

Hint: Here we will use the concept of splitting the middle terms and first making the pair of two terms and then finding the common factors from the paired terms and finally the factors for the given expression.

Complete step-by-step answer:
Take the given expression
 $ {a^2} - 2ab + {b^2} $
Now we will use the concept of to split the middle term.
Here we have three terms in the given expression.
Now, multiply the constant in the first term with the last term.
i.e. $ 1 \times (1) = (1) $
Now, you have to split the middle term to get $ (1) $ in multiplication and addition or subtraction to get the middle term i.e. $ ( - 2) $ . Here applying the basic concept of the product of two negative terms gives us the positive term and addition of two negative terms gives the value in the negative sign.
 $
  (1) = ( - 1) \times ( - 1) \\
  ( - 2) = ( - 1) - 1 \;
  $
Write the equivalent value for the middle term –
 $ {a^2} - ab - ab + {b^2} $
Now, make the pair of two terms in the above equation-
 $ = \underline {{a^2} - ab} \underline { - ab + {b^2}} $
Find the common factors from the paired terms –
 $ = a(a - b) - b(a - b) $
Take the common factors in the above equation –
 $ = (a - b)(a - b) $
By using the law of power and exponent, when bases are the same then powers are added.
 $ = {(a - b)^{1 + 1}} $
Simplify the above expression
 $ = {(a - b)^2} $
Hence, factor of $ {a^2} - 2ab + {b^2} $ is $ = {(a - b)^2} $
So, the correct answer is “ $ {(a - b)^2} $ ”.

Note: Here we were able to split the middle term and find the factors but in case it is not possible then we can find factors by using the formula\[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\] and considering the general form of the quadratic equation $ a{x^2} + bx + c = 0 $ . Be careful about the sign convention and simplification of the terms in the equation.