Problem 30 - Introductory Algebra for College Students

Question

Solve each quadratic equation by the square root property. If possible, simplify radicals or rationalize denominators. $$(z-6)^{2}=12$$

Step-by-Step Solution

Answer

The solutions for the equation are \(z=6+2\sqrt{3}\) and \(z=6-2\sqrt{3}\)

Step 1: Apply the Square Root Property

The given equation is \((z-6)^{2}=12\). The square root property implies that the square root of an equation equals the positive and negative square roots of their results. Therefore, we can say that \(z-6=\sqrt{12}\) or \(z-6=-\sqrt{12}\).

Step 2: Simplify the radicals

The square root of 12 can be calculated as \(\sqrt{4} \times \sqrt{3}\) which simplifies to \(2\sqrt{3}\). Then, our equations become \(z-6=2\sqrt{3}\) and \(z-6=-2\sqrt{3}\).

Step 3: Solve for z

To solve for z, we add 6 to both sides of equation for each case. It results in \(z=6+2\sqrt{3}\) and \(z=6-2\sqrt{3}\).

Key Concepts

Square Root Property

The square root property is a useful technique for solving certain quadratic equations. It allows us to take the square root of both sides of an equation. This can simplify the process, especially when dealing with perfect squares. For the equation \((z-6)^2 = 12\), we use the square root property to assert that:

• \(z-6 = \sqrt{12}\)
• or \(z-6 = -\sqrt{12}\)

By doing this, we recognize that a square can have both a positive and negative root.
This insight helps us to branch into two separate sets of equations to solve for \(z\). Always remember to apply both the positive and negative roots when using this property.

Simplifying Radicals

Simplifying radicals involves breaking down a radical into its simplest form. This can often make calculations more manageable and results clearer. When you encounter a number like \(\sqrt{12}\), it's beneficial to find any perfect square factors.
The number 12 can be split as \(4 \times 3\), where 4 is a perfect square. Hence, \(\sqrt{12} = \sqrt{4} \times \sqrt{3}\).

• \(\sqrt{4} = 2\)
• Thus, \(\sqrt{12} = 2\sqrt{3}\)

In our equation, this simplification helps us express \(z-6 = 2\sqrt{3}\) or \(z-6 = -2\sqrt{3}\). Always look for perfect square factors when simplifying radicals to make your equations tidier.

Rationalizing Denominators

Rationalizing denominators can often lead to a neater expression when dealing with fractions containing radicals. Although our specific problem didn't involve fractions, the general principle remains essential. Consider if you encounter \(\frac{1}{\sqrt{3}}\).
To rationalize the denominator, multiply both the numerator and denominator by \(\sqrt{3}\):

• This gives you \(\frac{\sqrt{3}}{3}\).

Rationalizing helps in eliminating radicals from the denominator, making expressions easier to handle.
While not used directly in the given exercise, it's a fundamental skill to apply when appropriate.