Simplify Algebraic Expression

We need to find the value of the expression $100\left[\sqrt{1 + 2(3a) + 9a^2} - 4a\right]$ given $a = 0.1125$.

First, focus on the term inside the square root:

$1 + 2(3a) + 9a^2 = 1 + 6a + 9a^2$

Notice that $1 + 6a + 9a^2$ is a perfect square trinomial. It can be factored as:

$1 + 6a + 9a^2 = (1 + 3a)^2$

Calculate Value Using Substitution

Substitute this back into the original expression:

$100\left[\sqrt{(1 + 3a)^2} - 4a\right]$

Since $a = 0.1125$, the value $1 + 3a = 1 + 3(0.1125) = 1 + 0.3375 = 1.3375$, which is positive. Therefore, the square root simplifies to:

$\sqrt{(1 + 3a)^2} = 1 + 3a$

The expression now becomes:

$100\left[(1 + 3a) - 4a\right]$

Simplify the terms inside the brackets:

$100\left[1 - a\right]$

Now substitute the value $a = 0.1125$:

$100\left[1 - 0.1125\right]$

Perform the subtraction:

$100\left[0.8875\right]$

Finally, perform the multiplication:

$88.75$