Problem 30 - Elementary Algebra

Question

\(z^{2}=17\)

Step-by-Step Solution

Answer

The solutions are \(z = \sqrt{17}\) and \(z = -\sqrt{17}\).

Step 1 - Identify the Type of Equation

Recognize that the given equation is a quadratic equation of the form \(z^2 = 17\).

Step 2 - Apply the Square Root Property

To solve for \(z\), take the square root of both sides of the equation. Remember to consider both the positive and negative roots. This gives \(z = \pm \sqrt{17}\).

Step 3 - Simplify the Solution

Simplify the expression for \(z\). The solutions are \(z = \sqrt{17}\) and \(z = -\sqrt{17}\).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation of degree 2, meaning the highest power of the variable is 2. A standard quadratic equation can be written in the form: \[ ax^2 + bx + c = 0 \] Here, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In the given exercise, the quadratic equation is \( z^2 = 17 \). This is a simplified form because there are no \( b \) and \( c \) terms. When solving quadratic equations, you need to find the value(s) of the variable that make the equation true. We can solve quadratic equations using a variety of methods including factoring, using the quadratic formula, completing the square, or applying the square root property.

Square Root Property

The square root property is a powerful tool for solving quadratic equations of the form \( x^2 = k \), where \( k \) is a constant. This property states that if \( x^2 = k \), then: \[ x = \pm \sqrt{k} \] This means that \( x \) can be either the positive or the negative square root of \( k \). Applying this to the given exercise \( z^2 = 17 \), we take the square root of both sides to get: \[ z = \pm \sqrt{17} \] This step is crucial as it helps to find both possible solutions for \( z \). Make sure not to forget the \( \pm \) symbol, which indicates both the positive and negative roots.

Positive and Negative Roots

When solving quadratic equations using the square root property, it's important to recognize that you will typically have two solutions: one positive and one negative. This is because squaring either a positive or a negative number results in a positive value. For instance, in the exercise: \[ z^2 = 17 \] Taking the square root of 17 gives us two solutions: \[ z = \sqrt{17} \] and \[ z = -\sqrt{17} \] These represent the positive and negative roots, respectively. During practice, always consider both roots to ensure you don't miss any solution. This concept is fundamental when solving quadratic equations, as it helps in finding all possible values that satisfy the equation.