91Ó°ÊÓ

In algebra, isolating the variable means rearranging the equation so that the variable you are solving for stands alone on one side of the equation.
This is a critical step because it allows you to focus on the term that contains the variable.

Let’s look at the initial equation: \[a^2 - 40 = 0\]
To isolate the term with the variable (\(a^2\)), we need to move the -40 to the other side.
We do this by adding 40 to both sides:
\[a^2 - 40 + 40 = 0 + 40\]
This simplifies to: \[a^2 = 40\]
Now, the squared term is isolated, which means it stands alone on one side of the equation.
This step is foundational in solving equations because it sets the stage for further operations like taking square roots.

Taking the square root is the process of finding a number that, when multiplied by itself, gives the original number.
When you have an equation like \[a^2 = 40\]
the next logical step is to take the square root of both sides to solve for \(a\).
Applying square roots to both sides, we get:
\[\begin{aligned} a &= \sqrt{40} \ &= \pm\sqrt{40} \end{aligned}\]
You'll notice that we write \(a = \pm\sqrt{40}\).
This is because both positive and negative values will satisfy the equation when squared.
In other words, both \(2\sqrt{10}\) and \(-2\sqrt{10}\) are valid solutions.
Hence, \(a = \, 2\sqrt{10}\) and \(a = \,-2\sqrt{10}\)'.

Simplifying radicals is the process of breaking down a radical expression into its simplest form.
It often involves finding the prime factors of the number under the radical sign.
In our case, we need to simplify \[\sqrt{40}\]
Let's break it down:
\[\begin{aligned} \sqrt{40} &= \sqrt{4 \cdot 10} \ &= \sqrt{4} \cdot \, \sqrt{10} \ &= 2\sqrt{10} \end{aligned}\]
You see, \[\sqrt{40} \= 2\sqrt{10}\]
We break 40 into 4 and 10 because 4 is a perfect square, and its square root is 2.
Multiplying by \[ \, \, \sqrt{10}\]
leaves us with \[ \, \sqrt{40} \ = 2\sqrt{10}\].
Therefore, the final solutions to the equation \[a = \pm 2\sqrt{10}\].