Question

How do you multiply (x + 8)(x - 8)?

Answer 1

Hint: Now to multiply the given expression we will use distributive property. Hence we will expand the expression 2 times by distributive property. Then we will simplify by adding and subtracting similar terms and hence we will get the multiplication of given expression.

Complete step by step solution:
Now let us first understand some basic properties in real numbers. Let a, b and c be any real numbers then,
Commutative property states that a + b = b + a similarly in multiplication we have a.b = b.a
Now associative property states that a + (b + c) = (a + b) + c similarly in multiplication we have (a.b).c = a.(b.c)
Now let us understand distributive property.
It states that a.(b + c) = ab + ac.
Now we will use some of these properties to solve the given expression.
Now let first consider the given expression (x + 8)(x - 8).
Now we know according to distributive property we have $a\left( b-c \right)=ab-ac$ .
Now here we will consider (x + 8) as one term and use the distributive property.
Hence we get, $\left( x+8 \right)\times x-\left( x+8 \right)\times 8$ .
Now again by distributive property we have $x\left( x+8 \right)=x\times x+8\times x$ and similarly we have $\left( x+8 \right)8=8\times x+8\times 8$ Hence substituting these in the expression we get.
$\begin{align}
  & \Rightarrow x\left( x \right)+x\left( 8 \right)-\left[ 8\left( x \right)+8\times 8 \right] \\
 & \Rightarrow {{x}^{2}}+8x-\left[ 8x+64 \right] \\
\end{align}$
Now opening the bracket of the above expression we get,
$\Rightarrow {{x}^{2}}+8x-8x-64$
Now we know that we can add and subtract the terms with the same degree hence we get 8x – 8x = 0. Hence we get the expression as,
$\Rightarrow {{x}^{2}}-64$
Hence the multiplication of (x + 8) and (x - 8) is ${{x}^{2}}-64$.

Note: Note that we can avoid the calculation and using the distributive property and still solve this expression. We know that $\left( {{a}^{2}}-{{b}^{2}} \right)=\left( a-b \right)\left( a+b \right)$ . Hence using the we can directly write $\left( x-8 \right)\left( x+8 \right)=\left( {{x}^{2}}-{{8}^{2}} \right)$ .