Quadratic equations represent a significant advancement in algebra, moving beyond linear equations to a more complex form. A quadratic equation is defined as a polynomial of degree 2, characterized by the presence of an \(x^2\) term. The standard form of a quadratic equation is expressed as:

\[ ax^2 + bx + c = 0 \]

In this equation, \(a\), \(b\), and \(c\) are coefficients, where \(a\) cannot be zero. The term \(ax^2\) is the leading term, followed by \(bx\) (which can be thought of as \(1x\) if the coefficient is not explicitly stated), and \(c\) is the constant term. The coefficients are identified as follows:

• a: The coefficient of the \(x^2\) term.
• b: The coefficient of the \(x\) term.
• c: The constant term.

For example, in the quadratic equation \(3x^2 + 2x - 6 = 0\), the coefficients are \(a = 3\), \(b = 2\), and \(c = -6\). It is essential to pay attention to the signs of the coefficients to ensure accurate identification.

To convert a non-standard equation into standard form, all terms must be moved to one side of the equation, ensuring they are arranged in descending order of power. For instance, to rewrite \(5x^2 = x - 3\) in standard form, you would subtract \(x\) and add \(3\) to both sides, resulting in:

\[ 5x^2 - x + 3 = 0 \]

Here, \(a = 5\), \(b = -1\), and \(c = 3\). In another example, the equation \(-2x^2 + \frac{5}{3} = 0\) is already in standard form, with \(a = -2\), \(b = 0\) (since there is no \(x\) term), and \(c = \frac{5}{3}\). This illustrates that \(b\) and \(c\) can indeed be zero or fractions, as long as the \(x^2\) term is present, confirming it as a quadratic equation.

Understanding how to manipulate and identify the components of quadratic equations is crucial for solving them, which will be explored in subsequent lessons.