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Factorize the following:

(i) [tex]$x^2 + 6xy + 9y^2$[/tex]


Sagot :

Sure, let's factorize the given expression step-by-step.

The given expression is:

[tex]\[ x^2 + 6xy + 9y^2 \][/tex]

Notice that the expression is a quadratic trinomial. To factorize it, we need to look for two binomials (in the form of [tex]\((ax + by)(cx + dy)\)[/tex]) whose product gives us the original quadratic trinomial.

1. Identify the first and the last term:
- The first term is [tex]\(x^2\)[/tex], which is [tex]\((x)(x)\)[/tex].
- The last term is [tex]\(9y^2\)[/tex], which is [tex]\((3y)(3y)\)[/tex].

2. Identify the middle term:
- The middle term is [tex]\(6xy\)[/tex].

3. Validate our factorization by checking the middle term:
- When we look at the possible binomials, we notice that if we use [tex]\((x + 3y)(x + 3y)\)[/tex], we need to check if this product gives us the correct middle term.

Now let's expand [tex]\((x + 3y)^2\)[/tex] to check if it matches the original expression:

[tex]\[ (x + 3y)(x + 3y) = x(x + 3y) + 3y(x + 3y) \][/tex]

4. Distribute [tex]\(x\)[/tex] and [tex]\(3y\)[/tex]:
- [tex]\(x(x + 3y) = x^2 + 3xy\)[/tex]
- [tex]\(3y(x + 3y) = 3xy + 9y^2\)[/tex]

5. Combine like terms:
- [tex]\(x^2 + 3xy + 3xy + 9y^2 = x^2 + 6xy + 9y^2\)[/tex]

We can see that the expansion of [tex]\((x + 3y)^2\)[/tex] gives us back the original expression [tex]\(x^2 + 6xy + 9y^2\)[/tex].

Hence, the factorized form of the given expression is:

[tex]\[ (x + 3y)^2 \][/tex]