Problem 62 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ (5 z+6)^{2}=75 $$

Step-by-Step Solution

Answer

z = \sqrt{3} - \frac{6}{5} or z = -\sqrt{3} - \frac{6}{5}

Step 1: Isolate the squared term

Start by isolating the term that is squared on one side of the equation: \( (5z + 6)^2 = 75 \)

Step 2: Apply the square root property

Apply the square root property by taking the square root of both sides of the equation: \( \sqrt{(5z + 6)^2} = \sqrt{75} \)This simplifies to: \( 5z + 6 = \pm \sqrt{75} \)

Step 3: Simplify the radical

Simplify the square root of 75: \( \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3} \)So the equation becomes: \( 5z + 6 = \pm 5\sqrt{3} \)

Step 4: Solve for z

Solve for \( z \) by isolating it: 1. \( 5z + 6 = 5\sqrt{3} \)Subtract 6 from both sides: \( 5z = 5\sqrt{3} - 6 \)Divide by 5: \( z = \frac{5\sqrt{3} - 6}{5} = \sqrt{3} - \frac{6}{5} \)2. \( 5z + 6 = -5\sqrt{3} \)Subtract 6 from both sides: \( 5z = -5\sqrt{3} - 6 \)Divide by 5: \( z = \frac{-5\sqrt{3} - 6}{5} = -\sqrt{3} - \frac{6}{5} \)

Key Concepts

Square Root Property

When an equation contains a squared term, the square root property is a straightforward method to solve it.
The property states that if you have something of the form \(x^2 = a\), then \(x = \pm \sqrt{a}\).
This means that \(x\) can be either the positive or negative square root of \(a\).
For instance, if we have the equation \((5z + 6)^2 = 75\), we apply the square root property by taking the square root of both sides.
This gives us \(5z + 6 = \pm \sqrt{75}\).
Remember, the \(\text{plus or minus}\) symbol indicates two possible values: one positive and one negative.

Simplifying Radicals

Radicals can often be simplified to make calculations easier.
To simplify \(\text{the square root of 75}\), we need to break it down into prime factors.
Notice that \(\text{75 = 25 * 3}\), so we can rewrite \(\text{the square root of 75}\) as \(\sqrt{25 \cdot 3}\).
Using the property of square roots, we get \(\sqrt{75} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}\).
This makes our equation \(5z + 6 = \pm 5\sqrt{3}\).
Simplifying radicals is crucial for reaching a manageable form of the equation.

Solving for Variables

To find the value of \(z\), we need to isolate it on one side of the equation.
Starting from our simplified radical equation, \(5z + 6 = \pm 5\sqrt{3}\), we follow these steps:

• Subtract 6 from both sides: \(5z = \pm 5\sqrt{3} - 6\)
• Then, divide both sides by 5: \z = \frac{5\sqrt{3} - 6}{5}\ or \(z = \frac{-5\sqrt{3} - 6}{5}\).

This results in two solutions: \(z = \sqrt{3} - \frac{6}{5}\) and \(z = -\sqrt{3} - \frac{6}{5}\).
Breaking down steps helps to avoid confusion and ensures correct answers.