Add to make Perfect Squares

Question

What must be added to each of the following expressions to obtain a perfect square?

[Q1] a^2 + 2.4a
[Q2] c^2 + (14/3)c
[Q3] v^2 – (7/2)v
[Q4] h^2 + 3mh
[Q5] k^2 – (4/3)k

Answer

Perfect squares have the ‘form’

(x + y)^2 = x^2 + 2xy + y^2

In the questions below, set x = a, c, v , h & k respectively for the questions 1-5 below. Then compare with the form above and solve for y!

[Q1] a^2 + 2.4a

(x + y)^2 =(a + y)^2 = a^2 + 2ay + y^2
So we have 2ay = 2.4a implying y = 1.2, meaning we just need to add ‘1.2^2’ to obtain a square. indeed,
a^2 + 2.4a + 1.2^2 = ( a + 1.2 )^2

[Q2] c^2 + (14/3)c

(x + y)^2 =(c + y)^2 = c^2 + 2cy + y^2
So we have 2cy = (14/3)c implying y = 7/3, meaning we just need to add ‘(7/3)^2’ to obtain a square. indeed,
c^2 + (14/3)c + (7/3)^2 = ( c + 7/3 )^2

[Q3] v^2 – (7/2)v

(v + y)^2 =(v + y)^2 = v^2 + 2vy + y^2
So we have 2vy = -(7/2)v implying y = -(7/4), meaning we just need to add ‘(-7/4)^2’ to obtain a square. indeed,
v^2 – (7/2)v + (-7/4)^2 = ( v – 7/4 )^2

[Q4] h^2 + 3mh

(h + y)^2 =(h + y)^2 = h^2 + 2hy + y^2
So we have 2hy = 3mh implying y = 1.5m, meaning we just need to add ‘(1.5m)^2’ to obtain a square. indeed,
h^2 + 3mh + (1.5m)^2 = ( h + 1.5m )^2

[Q5] k^2 – (4/3)k

(x + y)^2 =(k + y)^2 = k^2 + 2ky + y^2
So we have 2ky = -(4/3)k implying y = -(2/3)k, meaning we just need to add ‘(-2/3)^2’ to obtain a square. indeed,
k^2 – (4/3)k + (-2/3)^2 = ( k – 2/3 )^2

We are done.