How do you find it is a quadratic equation?

You can identify a quadratic equation by its form and key characteristics.

Here's how:

Defining Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually x) is 2. The standard form of a quadratic equation is:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0.

Identifying Characteristics

Here are the key characteristics to look for when determining if an equation is quadratic:

• Presence of an x² term: The equation must have a term where the variable (x) is raised to the power of 2. This is the defining feature.
• No higher powers: The equation should not contain any terms where the variable is raised to a power greater than 2 (e.g., x³, x⁴, etc.).
• Standard Form: The equation can be rearranged into the standard form ax² + bx + c = 0.

Examples

Let's look at some examples to illustrate how to identify quadratic equations:

• Example 1: Quadratic

  • 3x² + 2x - 1 = 0
  • This is a quadratic equation because it has an x² term, no higher powers of x, and is in the standard form.

• Example 2: Not Quadratic (Linear)

  • 2x + 5 = 0
  • This is not a quadratic equation because it does not have an x² term; it's a linear equation.

• Example 3: Not Quadratic (Cubic)

  • x³ - 4x² + x = 0
  • This is not a quadratic equation because it has an x³ term; it's a cubic equation.

Quadratic Functions and Graphs

While the question focuses on equations, it's worth noting that quadratic functions graph as parabolas. According to the reference, the vertex form of a quadratic equation can be used to find the equation from a graph:

y = a(x - h)² + k

Where (h, k) is the vertex of the parabola. If you know the vertex and one other point on the parabola, you can determine the value of a and thus define the specific quadratic equation.