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

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,

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.