Factor completly
9a^2+42a+49
The given expression 9a^2 + 42a + 49 is a perfect square trinomial. It can be factored completely as:
(3a + 7)^2
To factor completely, we need to find two binomials that, when multiplied, equal the given polynomial.
Let's start by looking for two numbers that multiply to give 9 * 49 = 441 (the product of the leading coefficient and the constant term) and add up to give 42 (the coefficient of the middle term).
The numbers 9 and 49 satisfy these conditions since 9 * 49 = 441 and 9 + 49 = 58.
Using these numbers, we can rewrite the middle term as 9a * 49a, which allows us to factor by grouping:
9a^2 + 42a + 49
= 9a^2 + 9a + 49a + 49
= (9a^2 + 9a) + (49a + 49)
= 9a(a + 1) + 49(a + 1)
Now we can factor out the common binomial factor, (a + 1):
= (a + 1)(9a + 49)
Therefore, the polynomial 9a^2 + 42a + 49 can be factored completely as (a + 1)(9a + 49).
To factor the expression 9a^2 + 42a + 49 completely, you can follow these steps:
Step 1: Check for a common factor, if any.
In this case, there is no common factor other than 1.
Step 2: Look for a pattern or method that can help you factor it.
The given expression is a perfect square trinomial. Perfect square trinomials can be factored using the formula (a + b)^2 = a^2 + 2ab + b^2.
Step 3: Apply the perfect square trinomial formula.
Using the formula (a + b)^2 = a^2 + 2ab + b^2, we can see that the expression 9a^2 + 42a + 49 matches the pattern of (a + b)^2, where a = 3a and b = 7.
So, we can write 9a^2 + 42a + 49 as (3a + 7)^2.
Therefore, the factored form of 9a^2 + 42a + 49 is (3a + 7)^2.