Question

What must be subtracted from \[3{a^2} - 6ab - 3{b^2} - 1\] to get \[4{a^2} - 7ab - 4{b^2}\].

Answer 1

Hint: Like terms are the terms having the same variables whereas unlike terms have different variables. For example, $4a$ and $6a$ are like terms because they have the same variable ‘$a$’. but $4{a^2}$ and $6a$ are unlike terms because they have different variables.
Addition is the summing of two or more than two numbers, or values, or variables and in addition if we sum the two or more numbers a new frame of number will be found, also in subtraction which is the minus of two or more than two numbers or values example $2 - 3 = - 1$ (larger number signs stays constant)

Complete step-by-step solution:
Let the required number be X.
According to the question,
\[3{a^2} - 6ab - 3{b^2} - 1 - (X)\] = \[4{a^2} - 7ab - 4{b^2}\]
On transposing the terms, the signs will be reversed, so
X =\[3{a^2} - 6ab - 3{b^2} - 1 - \]\[\left( {4{a^2} - 7ab - 4{b^2}} \right)\]
X = \[3{a^2} - 6ab - 3{b^2} - 1 - 4{a^2} + 7ab + 4{b^2}\]
X = \[ - {a^2} + ab + {b^2} - 1\]
So, X = \[ - {a^2} + ab + {b^2} - 1\] should be subtracted from \[3{a^2} - 6ab - 3{b^2} - 1\] to get \[4{a^2} - 7ab - 4{b^2}\].

Note: Whenever we move term from left hand side of equal to right hand side of equal to or vice versa then the signs are changed as-
+ into – and – into + (Addition into subtraction and vice versa.)
\[ \times \]into \[ \div \] and vice versa (Multiplication into division and vice versa).
Also, we have to know about the concept of multiplication and division.
Since multiplicand refers to the number multiplied. Also multiplier refers to the number multiplied by the first number. Have a look at an example; while multiplying $5 \times 7$ the number $5$ is called the multiplicand and the number $7$ is called the multiplier.
The process of the inverse of the multiplication method is called the division. Like $x \times y = z$ is multiplication thus the division sees as $x = \dfrac{z}{y}$. We usually need to memorize the multiplication tables in childhood so it will help to do mathematics.