Question

factor.
$64x^2 + 48x + 9$

Explanation:

Step1: Identify the form of the quadratic

The quadratic \(64x^2 + 48x + 9\) is a perfect square trinomial. A perfect square trinomial has the form \(a^2 + 2ab + b^2=(a + b)^2\).
Here, \(a^2 = 64x^2\), so \(a=\sqrt{64x^2}=8x\). And \(b^2 = 9\), so \(b=\sqrt{9}=3\).

Step2: Check the middle term

The middle term of a perfect square trinomial should be \(2ab\). Let's calculate \(2ab\) with \(a = 8x\) and \(b=3\): \(2\times(8x)\times3=48x\), which matches the middle term of the given quadratic.

Step3: Factor the quadratic

Using the perfect square trinomial formula \(a^2 + 2ab + b^2=(a + b)^2\), substitute \(a = 8x\) and \(b = 3\). So \(64x^2+48x + 9=(8x + 3)^2\).

Answer:

\((8x + 3)^2\)