What must be added to each of the following expressions to make it a whole square
(i) $4x^2 - 12x + 7$
(ii) $4x^2 - 20x + 20$

Given:

The given expressions are

(i) $4x^2 - 12x + 7$

(ii) $4x^2 - 20x + 20$

To do:

We have to find the term that must be added to each of the given expression to make it a whole square.

Solution:

The given expressions are (i) $4x^2 - 12x + 7$ (ii) $4x^2 - 20x + 20$. Here, we have to find the term that must be added to each of the given expression to make it a whole square. So, to find the term that must be added, we have to make the given expressions as the sum of a whole square and some other term and using the identities $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$, we can find the required values.

$(a+b)^2=a^2+2ab+b^2$.............(I)

$(a-b)^2=a^2-2ab+b^2$.............(II)

(i) The given expression is $4x^2 - 12x + 7$.

To make $4x^2 - 12x + 7$ as the sum of a whole square and some other term, add and subtract 9.

$4x^2 - 12x + 7=4x^2-12x+7+9-9$

$4x^2 - 12x + 7=(2x)^2-2(2x)(3)+(3)^2+7-9$

$4x^2 - 12x + 7=(2x-3)^2-2$                  [Using (II)]

Here, $(2x-3)^2-2+2=(2x-3)^2$ is a whole square.

Hence, $2$ must be added to the given expression to make it a whole square.

(ii) The given expression is $4x^2 - 20x + 20$

To make $4x^2 - 20x + 20$ as the sum of a whole square and some other term, add and subtract 25.

$4x^2 - 20x + 20=4x^2-20x+20+25-25$

$4x^2 - 20x + 20=(2x)^2-2(2x)(5)+(5)^2+20-25$

$4x^2 - 20x + 20=(2x-5)^2-5$                  [Using (II)]

Here, $(2x-5)^2-5+5=(2x-5)^2$ is a whole square.

Hence, $5$ must be added to the given expression to make it a whole square.

Akhileshwar Nani

e

Updated on: 04-Apr-2023

488 Views

Kickstart Your Career

Get certified by completing the course

Get Started