Question

Find the factors of the given quadratic expression ${x^2} + 6x + 9$.

Answer 1

Hint: Factors are numbers or algebraic expressions that divide another number or algebraic expression completely without leaving any remainder. To find the factors of a quadratic expression we have to reduce it to its factors by simply using factorization method.

Complete step by step answer:
Firstly we write down the expression given in the question-
$ \Rightarrow {x^2} + 6x + 9$
Now we want to find the factors of this expression so we use Factorization to reduce the expression to its factors and also keep in mind that factors of an algebraic expression are irreducible meaning that it cannot be reduced further.

So when an expression which has its highest power as 2 is given then this expression is quadratic so to find its factors we factorize it as products of its factors. We can use these methods to find the factors-
Grouping method: In this method we work upon the expression which are of this form ${x^2} + ax + b$ and then look for two factors $p\;{\text{and}}\;q$ such that
$ \Rightarrow p \times q = b\;{\text{and}}\;p + q = a$
Then the given expression can be written as
$ \Rightarrow {x^2} + (a + b)x + ab$
Now multiplying $x$ inside the bracket and then taking common terms together
$
 \Rightarrow {x^2} + ax + bx + ab \\
   \Rightarrow x(x + a) + b(x + a) \\
   \Rightarrow (x + a)(x + b) \\
 $
These two separate expressions are the factors of the given algebraic expression.
So by observing the given expression carefully we can say that we can have $p = 3\;{\text{and}}\;q = 3$ such that
$
 \Rightarrow 3 \times 3 = 9 = b \\
 \Rightarrow 3 + 3 = 6 = a \\
 $
Using the grouping method we have
\[
 \Rightarrow {x^2} + 6x + 9 \\
   \Rightarrow \;{x^2} + (3 + 3)x + 3 \times 3 \\
   \Rightarrow {x^2} + 3x + 3x + 9 \\
 \]
Taking common factors of first two and last two elements we have
$
 \Rightarrow {x^2} + 3x + 3x + 9 = x(x + 3) + 3(x + 3) \\
   \Rightarrow (x + 3)(x + 3) \\
 $
That’s it! We have two similar factors as $(x + 3)$ our answer.

Using identities: We simply make our expression like a standard algebraic identity and then check for the factors
Here the given expression ${x^2} + 6x + 9$ can be compared with the identity
$ \Rightarrow {x^2} + 2xy + {y^2} = {(x + y)^2}$
So we have from this
$
 \Rightarrow 2xy = 6x \\
  \Rightarrow y = 3 \\
 $
Now we substitute the values and write the expression
$ \Rightarrow {x^2} + 6x + 9 = {(x + 3)^2} = (x + 3)(x + 3)$
That’s it! We have two similar factors as $(x + 3)$ our answer.

Note: We can use both the methods for finding the factors of an algebraic expression. It is always convenient to use a second method when you cannot guess the two numbers to group the expression by observing the expression.