Question

What is a zero solution of an equation?

Answer 1

A **zero solution** of an equation refers to a value of the variable that makes the equation equal to zero. In simpler terms, it is the **value** or values of the variable(s) that satisfy the equation when the result is set to zero.

### Let's break this down further:

1. **General Equation Form**:
   An equation typically looks like this:
   
   \[
   f(x) = 0
   \]
   
   Here, \( f(x) \) represents some expression involving the variable \( x \). The goal is to find the value(s) of \( x \) that make \( f(x) \) equal to zero. This value is called a **zero solution** or simply a **zero**.

2. **Example of a Linear Equation**:
   Consider the equation:
   
   \[
   2x - 4 = 0
   \]
   
   To find the zero solution, solve for \( x \):
   
   \[
   2x = 4
   \]
   \[
   x = \frac{4}{2} = 2
   \]
   
   So, \( x = 2 \) is the zero solution of the equation because when you substitute \( x = 2 \) back into the equation, it makes the left-hand side \( 2(2) - 4 = 0 \).

3. **Example of a Quadratic Equation**:
   In a quadratic equation, you may have more than one zero solution. For example:
   
   \[
   x^2 - 5x + 6 = 0
   \]
   
   This equation can be factored as:
   
   \[
   (x - 2)(x - 3) = 0
   \]
   
   Here, two solutions exist:
   \[
   x = 2 \quad \text{and} \quad x = 3
   \]
   
   Both \( x = 2 \) and \( x = 3 \) are **zero solutions** because substituting either into the equation results in \( 0 \).

4. **Graphical Interpretation**:
   Graphically, a zero solution corresponds to the point(s) where the graph of the function crosses or touches the **x-axis**. These are called **x-intercepts**. For example, the graph of \( y = x^2 - 5x + 6 \) will cross the x-axis at \( x = 2 \) and \( x = 3 \).

5. **Other Types of Equations**:
   - In **polynomial equations**, zero solutions are often called **roots**.
   - In **trigonometric equations**, zero solutions may involve angles, such as solving \( \sin(x) = 0 \), which gives \( x = 0, \pi, 2\pi, \dots \).

### Key Points:
- A zero solution makes the equation equal to zero.
- It can be one or more values, depending on the type of equation.
- Graphically, zero solutions correspond to the x-intercepts of the graph of the function.

In summary, a zero solution is the value(s) of the variable that satisfy the equation when set equal to zero.

Answer 2

A "zero solution" of an equation refers to a specific value or values of the variable(s) that make the equation true, such that when you substitute these values into the equation, the result is zero. Essentially, it's where the function or expression represented by the equation equals zero.

Here's a detailed breakdown to clarify the concept:

1. **Definition of an Equation**: An equation is a mathematical statement that asserts the equality of two expressions. For example, in the equation \( f(x) = 0 \), \( f(x) \) is an expression that depends on \( x \).

2. **Zero Solution Concept**: A zero solution of an equation is a value of the variable that makes the equation true. Specifically, if you have an equation like \( f(x) = 0 \), a zero solution \( x_0 \) is a value where \( f(x_0) = 0 \). This means when \( x_0 \) is substituted into the expression \( f(x) \), the outcome is zero.

3. **Examples**:
    - **Linear Equation**: Consider the equation \( 2x - 4 = 0 \). To find the zero solution, solve for \( x \):
      \[
      2x - 4 = 0
      \]
      Add 4 to both sides:
      \[
      2x = 4
      \]
      Divide by 2:
      \[
      x = 2
      \]
      Here, \( x = 2 \) is the zero solution because substituting \( x = 2 \) into the original equation yields zero.

    - **Quadratic Equation**: For the quadratic equation \( x^2 - 5x + 6 = 0 \), find the zero solutions by factoring:
      \[
      (x - 2)(x - 3) = 0
      \]
      Setting each factor equal to zero gives:
      \[
      x - 2 = 0 \quad \text{or} \quad x - 3 = 0
      \]
      Thus, \( x = 2 \) and \( x = 3 \) are zero solutions, as substituting either value into the original equation results in zero.

4. **General Context**: The concept of zero solutions is fundamental in various mathematical contexts, including algebra, calculus, and differential equations. For instance, in calculus, finding the zeros of a function (where the function intersects the x-axis) is crucial for analyzing the behavior of the function.

In summary, a zero solution of an equation is a value or set of values for the variable(s) that satisfy the equation by making the expression on one side equal to zero.