91Ó°ÊÓ

A quadratic equation is a polynomial equation of the second degree. That means its highest exponent on the variable is two. Typically, these equations take the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). Quadratic equations can have real or complex solutions, and they are best visualized as parabolas when graphed on a coordinate plane.

Some key elements to keep in mind are:

• The quadratic term: \( ax^2 \), which defines the parabola's "width" and orientation.
• The linear term: \( bx \), which affects the parabola's direction.
• The constant term: \( c \), which determines the parabola's vertical position on the y-axis.

Solving quadratic equations usually involves finding values of \( x \) called the roots. These roots can be found using different methods, such as factoring, completing the square, or using the quadratic formula, which is a popular choice due to its direct application.

The discriminant is a key component of the quadratic formula. It's a part of the formula that helps determine the nature of the roots of the equation. It is expressed as \( D = b^2 - 4ac \). This value is crucial because it tells us:

• If \( D > 0 \): The equation has two distinct real roots.
• If \( D = 0 \): The equation has exactly one real root (also known as a repeated or double root).
• If \( D < 0 \): The equation has two complex roots, which means they are not real numbers.

Understanding the discriminant allows us to predict the number and type of solutions before we even solve the equation fully. In the given example, the discriminant was 49, indicating two distinct real roots.

The roots of a quadratic equation are the solutions for \( x \) in \( ax^2 + bx + c = 0 \). They are the values where the parabola crosses the x-axis on a graph. These can be found using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula channelizes the coefficients \( a \), \( b \), and \( c \) directly from the quadratic equation to find the roots.

Key steps involved include:

• Calculating the discriminant, \( D \).
• Finding the square root of \( D \).
• Plugging the values into the quadratic formula.
• Solving for the potential \( x \) values.

For the example equation \( 5x^2 + 3x - 2 = 0 \), using this method provides the roots \( x = \frac{2}{5} \) and \( x = -1 \), showing where the graph of the equation crosses the x-axis.