Question

Use suitable identities to find the following products:
(i) \[\left( {x + 4} \right)\left( {x + 10} \right)\]
(ii) \[\left( {x + 8} \right)\left( {x - 10} \right)\]
(iii) \[\left( {3x + 4} \right)\left( {3x - 5} \right)\]
(iv) \[\left( {{y^2} + \dfrac{3}{2}} \right)\left( {{y^2} - \dfrac{3}{2}} \right)\]
(v) \[\left( {3 - 2x} \right)\left( {3 + 2x} \right)\]

Answer 1

Hint:According to the given question, firstly observe the equation which is the form of \[\left( {x + a} \right)\left( {x + b} \right)\] or \[\left( {a + b} \right)\left( {a - b} \right)\]
Accordingly calculate the value of a and b respectively. Hence substitute the value in the formula that is \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\] and \[\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)\] .
Formula used:
 Here, we use the algebraic identities that is \[\left( {x + a} \right)\left( {x + b} \right)
= {x^2} + \left( {a + b} \right)x + ab\] and \[\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} -{b^2}} \right)\] .

Complete step by step solution:
(i) \[\left( {x + 4} \right)\left( {x + 10} \right)\]
Here, we will use the algebraic identity \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\]
Putting \[a = 4\] and \[b = 10\] .
On substituting the values in the identity we get,
\[ \Rightarrow \left( {x + 4} \right)\left( {x + 10} \right) = {x^2} + \left( {4 + 10} \right)x + 4 \times
10\]
On simplifying the right hand side we get,
\[ \Rightarrow \left( {x + 4} \right)\left( {x + 10} \right) = {x^2} + 14x + 40\]
(ii) \[\left( {x + 8} \right)\left( {x - 10} \right)\]
Here, we will use the algebraic identity \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a +
b} \right)x + ab\]
Putting \[a = 8\] and \[b = - 10\] .
On substituting the values in the identity we get,
\[ \Rightarrow \left( {x + 8} \right)\left( {x - 10} \right) = {x^2} + \left( {8 - 10} \right)x + 8 \times -10\]
On simplifying the right hand side we get,
\[ \Rightarrow \left( {x + 8} \right)\left( {x - 10} \right) = {x^2} - 2x - 80\]
(iii) \[\left( {3x + 4} \right)\left( {3x - 5} \right)\]
Taking out 3 common from the equation,
\[ \Rightarrow 3\left( {x + \dfrac{4}{3}} \right)\left( {x - \dfrac{5}{3}} \right)\]
\[ \Rightarrow 9\left( {x + \dfrac{4}{3}} \right)\left( {x - \dfrac{5}{3}} \right)\]
Here, we will use the algebraic identity \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\]
Putting \[a = \dfrac{4}{3}\] and \[b = \dfrac{{ - 5}}{3}\] .
On substituting the values in the identity we get,
\[ \Rightarrow \left( {3x + 4} \right)\left( {3x - 5} \right) = 9\left[ {{x^2} + \left( {\dfrac{4}{3} -
\dfrac{5}{3}} \right)x + \dfrac{4}{3} \times \dfrac{{ - 5}}{3}} \right]\]
After solving we get,
\[ \Rightarrow \left( {3x + 4} \right)\left( {3x - 5} \right) = 9\left[ {{x^2} + \left( {\dfrac{{ - 1}}{3}}
\right)x - \dfrac{{20}}{9}} \right]\]
Taking 9 inside the right hand side for multiplication,
\[ \Rightarrow \left( {3x + 4} \right)\left( {3x - 5} \right) = 9{x^2} - \left( {\dfrac{9}{3}} \right)x -\dfrac{{20}}{9} \times 9\]
On simplifying the right hand side we get,
\[ \Rightarrow \left( {3x + 4} \right)\left( {3x - 5} \right) = 9{x^2} - 3x - 20\]
(iv) \[\left( {{y^2} + \dfrac{3}{2}} \right)\left( {{y^2} - \dfrac{3}{2}} \right)\]
Here, we will use the algebraic identity \[\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} -
{b^2}} \right)\]
Putting \[a = {y^2}\] and \[b = \dfrac{3}{2}\] .
On substituting the values in the identity we get,
\[ \Rightarrow \left( {{y^2} + \dfrac{3}{2}} \right)\left( {{y^2} - \dfrac{3}{2}} \right) = {\left( {{y^2}}
\right)^2} - {\left( {\dfrac{3}{2}} \right)^2}\]
Now we will open the squares and simplify on right hand side we get,
\[ \Rightarrow \left( {{y^2} + \dfrac{3}{2}} \right)\left( {{y^2} - \dfrac{3}{2}} \right) = {y^4} -
\dfrac{9}{4}\]
(v) \[\left( {3 - 2x} \right)\left( {3 + 2x} \right)\]
Here, we will use the algebraic identity \[\left( {a + b} \right)\left( {a - b} \right) = \left( {{a^2} - {b^2}} \right)\]
Putting \[a = 3\] and \[b = 2x\] .
On substituting the values in the identity we get,
\[ \Rightarrow \left( {3 - 2x} \right)\left( {3 + 2x} \right) = {\left( 3 \right)^2} - {\left( {2x}
\right)^2}\]
Now we will open the squares and simplify on right hand side we get,
\[ \Rightarrow \left( {3 - 2x} \right)\left( {3 + 2x} \right) = 9 - 4{x^2}\]

Note: To solve these types of questions you must remember the algebraic identities and convert the equations according to the requirement of the identity. So, carefully observe the value of a and b while substituting in the formula.