Question

Use suitable identities to find the following products.
$\begin{align}
  & a)\left( x+8 \right)\left( x-10 \right) \\
 & b)\left( 3-2x \right)\left( 3+2x \right) \\
\end{align}$

Answer 1

Hint: In the given expressions we will use identities to find the product. Now for first expression we first convert the expression in form of (a – b) × (a + b) and use the identity $\left( x+a \right)\left[ \left( x-a \right)-b \right]=\left( x+a \right)\left( x-a \right)-b\left( x+a \right)$ to simplify. Now to solve the expression we will use the identity $\left( {{a}^{2}}-{{b}^{2}} \right)$ = (a – b) × (a + b).

Complete step by step answer:
Now first consider the expression (x + 8) × (x – 10).
Now let us write the equation as (x + 8) × (x – 8 – 2)
Now we know that $\left( x+a \right)\left[ \left( x-a \right)-b \right]=\left( x+a \right)\left( x-a \right)-b\left( x+a \right)$
Hence we have (x + 8) × (x – 10) = (x + 8) × (x – 8) – 2(x + 8)
We know that (a – b) × (a + b) = $\left( {{a}^{2}}-{{b}^{2}} \right)$
Here a = x and b = 8
Hence we have (x + 8) × (x – 10) = ${{x}^{2}}-{{8}^{2}}-2\left( x+8 \right)$
(x + 8) × (x – 10) = ${{x}^{2}}-64-2x-16$
(x + 8) × (x – 10) = ${{x}^{2}}-2x-80$
Hence the product of (x + 8) and (x – 10) is equal to ${{x}^{2}}-2x-80$
Now consider the equation (3 – 2x) × (3 + 2x).
Now the equation is in the form of (a – b) × (a + b) where a = 3 and b = 2.
We know that (a – b) × (a + b) = $\left( {{a}^{2}}-{{b}^{2}} \right)$
Hence we get (3 – 2x) × (3 + 2x) = ${{3}^{2}}-{{\left( 2x \right)}^{2}}$
(3 – 2x) × (3 + 2x) = $9-4{{x}^{2}}$
Hence the product of (3 – 2x) and (3 + 2x) is equal to $9-4{{x}^{2}}$

Note: Note that (3 – 2x) × (3 + 2x) = ${{3}^{2}}-{{\left( 2x \right)}^{2}}$ and not ${{3}^{2}}-2{{x}^{2}}$ . hence while using this expression take care that we square the whole second terms as well as first terms. Now we can derive the identity easily. First using distributive property we get
$\begin{align}
  & (a-b)\left( a+b \right)=a\left( a+b \right)-b\left( a+b \right) \\
 & \Rightarrow (a-b)\left( a+b \right)={{a}^{2}}+ab-ab-{{b}^{2}} \\
 & \therefore (a-b)\left( a+b \right)={{a}^{2}}-{{b}^{2}} \\
\end{align}$