completely factor the following expression. 16a^2-40ab+25b^2
Find two factors that multiply to get the end terms and add to the value of the center term. Since the center term is negative and the right term is positive, both of the second term factors must be negative.
(4a-5b)^2
To completely factor the expression 16a^2 - 40ab + 25b^2, we need to find two binomials that, when multiplied together, give us the original expression.
First, let's look at the expression 16a^2 - 40ab + 25b^2 and identify any factors that are common to all the terms. In this case, we can see that the expression is a perfect square trinomial because both the first and third terms (16a^2 and 25b^2) are perfect squares.
A perfect square trinomial can be factored as the square of a binomial. To factor it, we take the square root of the first term, the square root of the last term, and double the product of the square roots. In this case, we have:
√16a^2 = 4a (since 4^2 = 16 and √a^2 = a)
√25b^2 = 5b (since 5^2 = 25 and √b^2 = b)
Now, let's check if the sign and middle term match:
-40ab = -2 * (4a) * (5b)
The expression can be factored as:
(4a - 5b)(4a - 5b)
So, the completely factored form of the expression 16a^2 - 40ab + 25b^2 is (4a - 5b)(4a - 5b).