Problem 58 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ (7 z-5)^{2}=25 $$

Step-by-Step Solution

Answer

The solutions are \( z = \frac{10}{7} \) and \( z = 0 \).

Step 1: Isolate the squared term

The given equation is o (7z - 5)^2 = 25 The squared term is already isolated in this equation.

Step 2: Apply the square root property

Take the square root of both sides of the equation: \( \sqrt{(7z - 5)^2} = \sqrt{25} \) This simplifies to \(7z - 5 = \pm 5 \)

Step 3: Solve for both possible values

Consider both positive and negative cases: Case 1: \( 7z - 5 = 5 \) Add 5 to both sides: \(7z = 10 \) Divide by 7: \( z = \frac{10}{7} \) Case 2: \( 7z - 5 = -5 \) Add 5 to both sides: \( 7z = 0 \) Divide by 7: \( z = 0 \)

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations that can be written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. These equations are important because they appear in various real-world applications, such as physics and engineering. The equation given in the exercise, \( (7z - 5)^2 = 25 \), is a simplified form of a quadratic equation. To solve it, one effective method is to use the square root property.
The basic idea behind the square root property is that if \( x^2 = k \), then \( x = \pm \sqrt{k} \). Applying this to the equation \( (7z - 5)^2 = 25 \), we can solve for \( z \):

• First, ensure the squared term is isolated, which is already done in this case.
• Take the square root of both sides to get rid of the square: \( 7z - 5 = \pm 5 \).

Following these simple steps helps simplify the problem and makes it easier to solve for the variable.

Simplifying Radicals

Simplifying radicals means reducing them to their simplest form. Radicals often appear when solving quadratic equations, especially when using the square root property. In the given problem, we need to simplify \( \sqrt{25} \). Since 25 is a perfect square, its square root is straightforward: \( \sqrt{25} = 5 \).
However, if the radical is not a perfect square, you can still simplify it by factoring out the perfect squares. For example, \( \sqrt{50} \) can be broken down to \( \sqrt{25 \cdot 2} = 5 \sqrt{2} \). Here are some key points to remember:

• Look for perfect square factors in the radicand (the number inside the root).
• Factor out the perfect squares and take their square roots.
• Simplify the remaining radical if possible.

Simplifying radicals makes your final answers cleaner and often easier to understand.

Step-by-Step Algebra

Breaking down problems into smaller steps is vital for understanding and solving algebraic equations. In the exercise, we used a systematic approach to solve the quadratic equation. Here’s a detailed look at the process again:
1. **Isolate the Squared Term:** The equation \( (7z - 5)^2 = 25 \) already has the squared term isolated.
2. **Apply the Square Root Property:** Take the square root of both sides, which gives \( 7z - 5 = \pm 5 \).
3. **Solve for the Variable:** Consider both positive and negative cases:

• Case 1: \( 7z - 5 = 5 \) leads to \( 7z = 10 \) and thus \( z = \frac{10}{7} \).
• Case 2: \( 7z - 5 = -5 \) leads to \( 7z = 0 \) and thus \( z = 0 \).

By breaking the problem down into these steps, you can clearly see how to arrive at the solution efficiently. This method not only helps in solving this particular problem but also builds a framework for tackling other algebraic problems.