Question

factor completely. 25d² + 10d + 1

Explanation:

Step1: Identify the form

The quadratic expression \(25d^{2}+10d + 1\) is a perfect square trinomial. The general form of a perfect square trinomial is \(a^{2}+2ab + b^{2}=(a + b)^{2}\).
Here, \(a^{2}=25d^{2}\), so \(a = 5d\) (since \((5d)^{2}=25d^{2}\)). And \(b^{2}=1\), so \(b = 1\) (since \(1^{2}=1\)).

Step2: Check the middle term

Now, check the middle term \(2ab\). Substitute \(a = 5d\) and \(b = 1\) into \(2ab\), we get \(2\times(5d)\times1=10d\), which matches the middle term of the given expression.

Step3: Factor the expression

Using the perfect square trinomial formula \(a^{2}+2ab + b^{2}=(a + b)^{2}\), with \(a = 5d\) and \(b = 1\), we factor \(25d^{2}+10d + 1\) as \((5d + 1)^{2}\).

Answer:

\((5d + 1)^{2}\)