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The square root property is a vital concept when dealing with quadratic equations. It helps find solutions when an equation can be transformed into the form \( z^2 = a \). The principle here is straightforward: if the square of a variable equals some number, then the variable is the positive or negative square root of that number.

This property states: if \( z^2 = a \), then \( z = \pm \sqrt{a} \). Essentially, it tells us there are always two possible solutions: one positive and one negative. This approach relies on the fact that squaring either a positive or negative number results in the same positive number.

For instance, in the equation \( z^2 = 10 \), the solutions are \( z = \sqrt{10} \) and \( z = -\sqrt{10} \). It's essential to remember both possible values when solving quadratic equations using this property. The use of the square root property simplifies the process of finding solutions.

Before using the square root property, it's crucial to isolate the quadratic term. This makes it easier to handle the equation and apply further operations.

In the given problem, we start with the equation \( 3z^2 - 30 = 0 \). The goal is to have the variable \( z^2 \) by itself on one side of the equation. This is done by performing basic arithmetic operations. First, add 30 to both sides to eliminate the constant term:

• \( 3z^2 - 30 + 30 = 0 + 30 \)
• Which simplifies to \( 3z^2 = 30 \)

After getting rid of the constant term, the next step is to divide by the coefficient of \( z^2 \), which is 3 in this case. This operation ensures that the term \( z^2 \) stands alone:

• \( \frac{3z^2}{3} = \frac{30}{3} \)
• Resulting in \( z^2 = 10 \)

Isolating the quadratic term is a foundational step that simplifies solving the equation.

Solving equations involves a series of logical steps to find an unknown value. In the context of quadratic equations like \( 3z^2 - 30 = 0 \), the process includes isolating terms and using the right properties to find solutions.

First, ensure the equation is manageable by isolating any quadratic term, as discussed earlier. This allows us to transform the equation into a simpler form like \( z^2 = 10 \).

Once isolated, use the square root property to derive possible solutions. In our example:

• By applying the square root property to \( z^2 = 10 \), we obtain:
• \( z = \sqrt{10} \) and \( z = -\sqrt{10} \)

These two values encompass all possible solutions for the quadratic equation. Techniques such as isolating terms and proper mathematical properties help break down complex equations into simple steps, ensuring that every solution is both comprehensible and achievable.