Solve for Z Z^2=49

Solution:

Step 1:

Isolate the variable by applying the square root to both sides of the equation. $Z = \pm \sqrt{49}$

Step 2:

Find the square root of 49.

Step 2.1:

Express 49 as a square of an integer. $Z = \pm \sqrt{7^{2}}$

Step 2.2:

Extract the square root of the perfect square. $Z = \pm 7$

Step 3:

Determine both solutions from the positive and negative roots.

Step 3.1:

For the positive root: $Z = 7$

Step 3.2:

For the negative root: $Z = -7$

Step 3.3:

Combine both solutions to present the final answer. $Z = 7 , -7$

Knowledge Notes:

To solve an equation where the variable is squared, one typically takes the square root of both sides. This process is based on the principle that if $a^2 = b$, then $a = \pm \sqrt{b}$, where $\pm$ indicates that both the positive and negative square roots should be considered.

When simplifying the square root of a perfect square, such as $49$, it's helpful to recognize that $49$ is $7^2$. Thus, the square root of $49$ is $7$, and because of the $\pm$ symbol, we consider both $7$ and $-7$.

The final solution to an equation with a squared variable will often have two possible values, one positive and one negative, unless the context of the problem restricts the solution to only positive or only negative values.

In this case, since there are no such restrictions, both $7$ and $-7$ are valid solutions for $Z$ when $Z^2 = 49$.