91Ó°ÊÓ

A quadratic equation is a second-degree polynomial equation in a single variable, which can be written in the standard form as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The variable \( x \) represents the unknown that we are trying to solve for.
These equations are called 'quadratic' because they're related to a square (quadratus in Latin). Quadratic equations often appear in algebra, and they can represent various real-world scenarios, such as projectile motion or optimization problems.
To solve a quadratic equation, one can use methods like factoring, completing the square, or the quadratic formula. The Quadratic Formula is a universally applicable method for finding the roots of any quadratic equation, written as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula provides the solution to the equation by taking into account the coefficients \( a \), \( b \), and \( c \). It gives us two potential solutions due to the "\( \pm \)" symbol, which indicates that there can be two values of \( x \). The use of such a formula makes solving any quadratic equation straightforward and consistent.

The discriminant is a crucial part of solving quadratic equations, found in the quadratic formula under the square root sign: \( \sqrt{b^2 - 4ac} \). This expression, \( b^2 - 4ac \), is called the discriminant and is denoted by \( D \). The value of \( D \) helps determine the nature of the roots of a quadratic equation.
Depending on the value of the discriminant, you can predict:

• **Positive Discriminant**: If \( D > 0 \), the quadratic equation has two distinct real roots.
• **Zero Discriminant**: If \( D = 0 \), the quadratic equation has exactly one real root, or in other words, two identical real roots (also called repeated roots).
• **Negative Discriminant**: If \( D < 0 \), the quadratic equation has two complex roots, which are complex conjugates of each other.

When solving the equation \( x^2 + 6x + 10 = 0 \), the discriminant is calculated as \( -4 \), a negative number, indicating that the solutions include complex roots.

When a quadratic equation has a negative discriminant, it means the equation has complex roots. Complex numbers are numbers that have a real part and an imaginary part. They are expressed in the form \( a + bi \), where \( i \) is the imaginary unit, defined by \( i^2 = -1 \).
In our example equation, \( x^2 + 6x + 10 = 0 \), we found the discriminant to be \( -4 \). As seen in the solution process, a negative discriminant results in an expression under the square root that involves an imaginary number. The equation's roots are calculated using:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Plugging in the values, the roots become:

• \( x = \frac{-6 \pm \sqrt{-4}}{2} = -3 \pm i \)

This indicates the equation has two complex roots: \( x = -3 + i \) and \( x = -3 - i \). Complex roots in pairs like these are common when dealing with quadratic equations that have a negative discriminant.