91Ó°ÊÓ

Quadratic equations are mathematical expressions that take the form \( ax^2 + bx + c = 0 \). Here, \( x \) is the variable, and \( a \), \( b \), and \( c \) are coefficients, with \( a eq 0 \). This is because if \( a \) were zero, we would have a linear equation, not quadratic.
It’s essential to understand that a quadratic equation is a polynomial of degree two. In simple terms, the highest power of the variable \( x \) is two. Examples of quadratic equations include \( x^2 + 5x + 6 = 0 \), \( 2x^2 - 4x - 3 = 0 \), and \( x^2 - x - 2 = 0 \).

Solving quadratic equations can involve several methods:

• Factoring
• Completing the square
• Using the quadratic formula

In our exercise, we use the quadratic formula, which is handy when factoring is complex or not immediately possible. The quadratic formula photogenically uncovers the roots of any quadratic equation:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). These roots can be real or complex numbers.

Complex numbers come into play when dealing with quadratic equations that do not have real number solutions. A complex number is expressed in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part.

The term imaginary might be misleading; these numbers are very much real within the mathematical space. The key component \( i \) is defined as \( \sqrt{-1} \). This is why, when the discriminant (\( b^2 - 4ac \)) is negative, we express the solution using complex numbers.

In the original exercise, after calculating the discriminant to be \(-2\), we recognize that \( \sqrt{-2} \) doesn't have a real solution. This leads us to use \( i \) to rewrite \( \sqrt{-2} \) as \( i\sqrt{2} \). Thus, complex solutions tell us about the roots of the quadratic equation, even when those roots can’t be plotted on the real number line.

The discriminant of a quadratic equation is the expression \( b^2 - 4ac \) found under the square root symbol in the quadratic formula. The discriminant gives valuable information about the nature of the roots.

Here’s what you need to know:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is one real root (or a repeated root).
• If the discriminant is negative, the equation has complex roots.

In our exercise, the discriminant \( b^2 - 4ac = -2 \) was negative. This immediately signals that the solutions will be complex numbers, hence using \( i \). In this case, the roots were found to be \( x = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \) and \( x = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \).
The discriminant plays a crucial role in predicting the type of solutions without solving the entire equation.