Question

Expand:
A) ${\left( {3a + 2b} \right)^2}$
B) ${\left( {5x - 7y} \right)^2}$

Answer 1

Hint: Here we use the formula of algebraic expansion. The formulas of algebraic expansion are ${\left( {a + b} \right)^2} = {\left( a \right)^2} + {\left( b \right)^2} + 2\left( a \right)\left( b \right)$ and ${\left( {a - b} \right)^2} = {\left( a \right)^2} + {\left( b \right)^2} - 2\left( a \right)\left( b \right)$. Now, we calculate the expansion of ${\left( {3a + 2b} \right)^2}$ and ${\left( {5x - 7y} \right)^2}$.

Complete step-by-step answer:
From the given data, to expand the equation ${\left( {3a + 2b} \right)^2}$ and ${\left( {5x - 7y} \right)^2}$ by using the above information.
A) To expand the given equation ${\left( {3a + 2b} \right)^2}$.
Here we use the algebraic formula of ${\left( {a + b} \right)^2} = {\left( a \right)^2} + {\left( b \right)^2} + 2\left( a \right)\left( b \right)$. Now, we compare the equation ${\left( {3a + 2b} \right)^2}$ with the algebraic expression ${\left( {a + b} \right)^2}$. Where, $a = 3a{\rm{ and }}b = 2b$.
Now, we substitute the value of a as 3a and b as 2b in the algebraic expression ${\left( {a + b} \right)^2} = {\left( a \right)^2} + {\left( b \right)^2} + 2\left( a \right)\left( b \right)$.
${\left( {3a + 2b} \right)^2} = {\left( {3a} \right)^2} + {\left( {2b} \right)^2} + 2\left( {3a} \right)\left( {2b} \right)\\
 = 9{a^2} + 4{b^2} + 2\left( {6ab} \right)\\
 = 9{a^2} + 4{b^2} + 12ab$
Hence, the expansion of the equation ${\left( {3a + 2b} \right)^2}$ is $9{a^2} + 4{b^2} + 12ab$.

B) To expand the given equation ${\left( {5x - 7y} \right)^2}$.
Here we use the algebraic formula of ${\left( {a - b} \right)^2} = {\left( a \right)^2} + {\left( b \right)^2} - 2\left( a \right)\left( b \right)$. Now, we compare the equation ${\left( {5x - 7y} \right)^2}$ with the algebraic expression ${\left( {a - b} \right)^2}$. Where, $a = 5x{\rm{ and }}b = - 7y$.
Again, we substitute the value of a as 5x and b as $ - 7y$ in the algebraic expression ${\left( {a - b} \right)^2} = {\left( a \right)^2} + {\left( b \right)^2} - 2\left( a \right)\left( b \right)$.
${\left( {5x - 7y} \right)^2} = {\left( {5x} \right)^2} + {\left( { - 7y} \right)^2} - 2\left( {5x} \right)\left( { - 7y} \right)\\
 = 25{x^2} + 49{y^2} - 2\left( { - 35xy} \right)\\
 = 25{x^2} + 49{y^2} + 70xy$
Hence, the expansion of the equation ${\left( {5x - 7y} \right)^2}$ is $25{x^2} + 49{y^2} + 70xy$.

Note: Here if we do not remember the formula of ${\left( {a + b} \right)^2} = {\left( a \right)^2} + {\left( b \right)^2} + 2\left( a \right)\left( b \right)$ and ${\left( {a - b} \right)^2} = {\left( a \right)^2} + {\left( b \right)^2} - 2\left( a \right)\left( b \right)$ then we simply use the multiplication method. Such as ${\left( {a + b} \right)^2} = \left( {a + b} \right) \times \left( {a + b} \right)$ and ${\left( {a - b} \right)^2} = \left( {a - b} \right) \times \left( {a - b} \right)$.