Question

If \[A = 4{x^2} + {y^2} - 6xy\]
\[B = 3{y^2} + 12{x^2} + 8xy\]
\[C = 6{x^2} + 8{y^2} + 6xy\]
Find (i) \[A + B + C\] (ii) \[\left( {A - B} \right) - C\]

Answer 1

Hint: Here, the value of each letter is given in the form of a quadratic equation. We will substitute all the values in the expression and perform the stated operation. In the first part we will add all the numbers and find the solution. In the second part, we will first subtract the terms in the bracket then subtract the value of C from the obtained term.

Complete step-by-step answer:
i.\[A + B + C\]
Here, substituting the values of A, B and C from the question, we get,
\[A + B + C = \left( {4{x^2} + {y^2} - 6xy} \right) + \left( {3{y^2} + 12{x^2} + 8xy} \right) + \left( {6{x^2} + 8{y^2} + 6xy} \right)\]
Now, adding the coefficients of same variables, we get,
\[ \Rightarrow A + B + C = \left( {4{x^2} + 12{x^2} + 6{x^2}} \right) + \left( {{y^2} + 3{y^2} + 8{y^2}} \right) + \left( {8xy - 6xy + 6xy} \right)\]
\[ \Rightarrow A + B + C = 22{x^2} + 12{y^2} + 8xy\]
Therefore, \[A + B + C = 22{x^2} + 12{y^2} + 8xy\]

ii. \[(A - B) - C\]
Substituting the values of A, B and C from the question, we get,
\[\left( {A - B} \right) - C = \left[ {\left( {4{x^2} + {y^2} - 6xy} \right) - \left( {3{y^2} + 12{x^2} + 8xy} \right)} \right] - \left( {6{x^2} + 8{y^2} + 6xy} \right)\]
Now, first we would solve (A\[ - \]B) as they are in the brackets.
\[ \Rightarrow \left( {A - B} \right) - C = \left[ {4{x^2} + {y^2} - 6xy - 3{y^2} - 12{x^2} - 8xy} \right] - \left( {6{x^2} + 8{y^2} + 6xy} \right)\]
(Due to ‘minus’ sign outside the bracket, all the ‘plus’ signs inside the second bracket were changed)
Now, solving further, we get,
 \[ \Rightarrow \left( {A - B} \right) - C = \left[ { - 8{x^2} - 2{y^2} - 14xy} \right] - \left( {6{x^2} + 8{y^2} + 6xy} \right)\]
Now, we will open the brackets and again, all the signs inside the second bracket will change due to the minus sign outside the bracket.
\[ \Rightarrow \left( {A - B} \right) - C = \left( { - 8{x^2} - 2{y^2} - 14xy - 6{x^2} - 8{y^2} - 6xy} \right)\]
\[ \Rightarrow \left( {A - B} \right) - C = - 14{x^2} - 10{y^2} - 20xy\]
Therefore,
 \[ \Rightarrow \left( {A - B} \right) - C = - 14{x^2} - 10{y^2} - 20xy\]


Note: The given terms have values in the form of a quadratic equation. A quadratic equation is a type of equation which has the highest degree of variable as 2. Here, we can only perform addition and subtraction on like terms. For example, the coefficients of \[{x^2}\] are added or subtracted together whereas the coefficient of \[x\] is added or subtracted together. We cannot add or subtract coefficients of \[{x^2}\] to the coefficient of \[x\], as they are two different entities. If we try to add them we will get some absurd answers. So we need to be careful while adding or subtracting the coefficient.