Question

What is the formula of $a$ plus $b$ minus $c$ the whole square?

Answer 1

Hint: We can recall that a trinomial is an algebraic expression composed of three terms connected by addition or subtraction. Here, we need to find the formula of $a$ plus $b$ minus $c$ the whole square. For this, we just need to multiply \[\left( {a + b - c} \right)\] by itself, we can easily derive the \[{\left( {a + b - c} \right)^2}\] formula. Thus, by simplifying this, we will get the final output.

Complete step by step answer:
Given that, we need to find the formula of a plus b minus c the whole square as below:
Let us see the expansion of \[{\left( {a + b - c} \right)^2}\] formula.
\[{(a + b - c)^2}= (a + b - c)(a + b - c)\]
For this, we select a term, and multiply it with each term in the other bracket, and then repeat until all terms have been selected.
\[{(a + b - c)^2} = {a^2} + ab - ac + ab + {b^{2\;}} - bc - ca - bc + {c^2}\]
\[\Rightarrow {(a + b - c)^2} = {a^2} + {b^{2\;}} + {c^2} + ab - ac + ab - bc - ca - bc\]
\[ \Rightarrow {(a + b - c)^2}= {a^2} + {b^2} + {c^2} + 2ab - 2bc - 2ca\]
\[\Rightarrow {(a + b - c)^2} = {a^2} + {b^2} + {c^2}\; + 2\left( {ab - bc - ca} \right)\]

Another Method: We know that,
\[{(a + b + c)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca\]
Thus, using this we can find the formula of
\[{(a + b - c)^2}= {(a + b + ( - c))^2}\]
\[\Rightarrow {(a + b - c)^2} = {a^2} + {b^2} + {( - c)^2} + 2ab + 2b( - c) + 2( - c)a\]
\[\Rightarrow {(a + b - c)^2}= {a^2} + {b^2} + {c^2} + 2ab - 2bc - 2ca\]

To check if the formula is correct or not, we will substitute some values of a, b and c.
Let, a=1, b=3, c=2
First,
\[ LHS= {(a + b - c)^2}\]
\[\Rightarrow LHS= {(1 + 3 - 2)^2}\]
\[\Rightarrow LHS= {(4 - 2)^2}\]
\[\Rightarrow LHS = {(2)^2}\]
\[\Rightarrow LHS= 4\]
And,
\[RHS = {a^2} + {b^2} + {c^2} + 2ab - 2bc - 2ca\]
\[\Rightarrow RHS = {(1)^2} + {(3)^2} + {(2)^2} + 2(1)(3) - 2(3)(2) - 2(2)(1)\]
\[\Rightarrow RHS = 1 + 9 + 4 + 6 - 12 - 4\]
\[\Rightarrow RHS = 20 - 16\]
\[\Rightarrow RHS = 4\]
Thus, LHS = RHS.

Hence, the formula of \[{(a + b - c)^2} = {a^2} + {b^2} + {c^2}\; + 2\left( {ab - bc - ca} \right)\].

Note: The square of the sum of three or more terms can be determined by the formula of the determination of the square of sum of two terms. Therefore, a perfect square trinomial can be defined as an expression that is obtained by squaring a binomial, which is an expression composed of two terms. If the first and last terms are perfect squares, and the middle term’s coefficient is twice the product of the square roots of the first and last terms, then the expression is a perfect square trinomial.