Expanding Perfect Squares
Just like perfect square numbers $9$9$\left(3^2\right)$(32) and $144$144$\left(12^2\right)$(122), algebraic expressions such as $a^2$a2, $x^2$x2 and $a^2b^2$a2b2 are also called perfect squares. Squares of binomial expressions, such as $\left(a+b\right)^2$(a+b)2, are also perfect squares, and we can expand these binomial products in the following way:
| $\left(a+b\right)^2$(a+b)2 | $=$= | $\left(a+b\right)\left(a+b\right)$(a+b)(a+b) | |
| $=$= | $a^2+ab+ba+b^2$a2+ab+ba+b2 | ||
| $=$= | $a^2+2ab+b^2$a2+2ab+b2 | Since $ab=ba$ab=ba |
$\left(x+y\right)^2=x^2+2xy+y^2$(x+y)2=x2+2xy+y2
The square of a binomial appears often, not just in maths but in the real world as well.
Squares like these can also be used to show the possible ways that genes can combine in offspring. For example, among tigers, the normal colour gene C is dominant, while the white colour gene c is recessive. So a tiger with colour genes of CC or Cc will have a normal skin colour, while a tiger with colour genes of cc will have a white skin colour. The following square shows all four possible combinations of these genes.
Since the square for each gene combination represents $\frac{1}{4}$14 of the area of the larger square, the probability that a tiger has colour genes of CC or Cc (i.e. it has a normal skin colour) is $\frac{3}{4}$34, while the probability that it has colour genes of cc (i.e. it has a white skin colour) is $\frac{1}{4}$14.
| $\left(\frac{1}{2}C+\frac{1}{2}c\right)^2$(12C+12c)2 | $=$= | $\frac{1}{4}C^2+2\left(\frac{1}{4}C\right)\left(\frac{1}{4}c\right)+\frac{1}{4}c^2$14C2+2(14C)(14c)+14c2 |
| $=$= | $\frac{1}{4}C^2+\frac{1}{2}Cc+\frac{1}{4}c^2$14C2+12Cc+14c2 |
More examples
Question 1
Complete the expansion of the perfect square: $\left(x-3\right)^2$(x−3)2
$\left(x-3\right)^2=x^2-\editable{}x+\editable{}$(x−3)2=x2−x+
Question 2
Write the perfect square trinomial that factorises as $\left(s+4t\right)^2$(s+4t)2.
Question 3
Expand the following perfect square: $\left(4x+7y\right)^2$(4x+7y)2