Question

How do you find the product ${(x + 3)^2}$?

Answer 1

Hint: The Square of a number or expression is the result that we get when the number or expression is multiplied by itself. For multiplying two expressions or an expression by itself we use the distributive property of multiplication, i.e. multiplying each term of an expression by each term of the other expression.

Complete step-by-step solution:
We have to find the product ${(x + 3)^2}$.
We can see that we have to find the square of the expression $(x + 3)$. In other words, we have to find the product of $(x + 3)$ multiplied by itself, i.e. we have to find,
$(x + 3) \times (x + 3)$
For this we will use the distributive property of multiplication wherein we multiply each term of an expression by each term of the other expression. This is given by,
$
 \Rightarrow (ax + b) \times (cx + d) \\
  \Rightarrow (ax \times (cx + d)) + (b \times (cx + d)) \\
   \Rightarrow (ax \times cx) + (ax \times d) + (b \times cx) + (b \times d) \\
$
Thus, we can find the product of $(x + 3) \times (x + 3)$ as,
$
 \Rightarrow (x + 3) \times (x + 3) \\
  \Rightarrow (x \times (x + 3)) + (3 \times (x + 3)) \\
  \Rightarrow (x \times x) + (x \times 3) + (3 \times x) + (3 \times 3) \\
   \Rightarrow {x^2} + 3x + 3x + 9 \\
    \Rightarrow {x^2} + 6x + 9 \\
$
We get the resulting product as $({x^2} + 6x + 9)$
Alternatively, we can also find the product of expression having two terms using the formula,
${(a + b)^2} = {a^2} + 2ab + {b^2}$
Putting $a = x$ and $b = 3$ we get,
$
 \Rightarrow {(x + 3)^2} \\
 \Rightarrow {x^2} + 2 \times x \times 3 + {3^2} \\
  \Rightarrow {x^2} + 6x + 9 \\
$

Hence, the product of ${(x + 3)^2}$ is $({x^2} + 6x + 9)$


Note: To find the square of an expression we multiply the expression by itself. For multiplication we can use the distributive property of multiplication wherein we multiply each term of an expression by each term of the other expression. For finding the square of an expression with two terms we can also use the direct formula given by, ${(a + b)^2} = {a^2} + 2ab + {b^2}$. The degree of the resulting expression that we get by squaring an expression is twice the degree of the given expression.