91Ó°ÊÓ

Quadratic equations are fundamental in algebra and represent polynomials of the second degree, typically in the form of \( ax^2 + bx + c = 0 \) where \( a \) , \( b \) , and \( c \) are constants and \( a \) is not equal to zero. To solve these equations, there are various methods, including factoring, completing the square, and using the quadratic formula, which is a reliable and universally applicable method regardless of the nature of the equation's roots.

The quadratic formula, \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \), is derived from the process of completing the square and provides an algebraic solution for the roots of any quadratic equation. When using this formula, it's essential to correctly identify the coefficients \( a \) , \( b \) , and \( c \) from the equation and substitute them accurately into the formula to find the values of \( x \), which are the solutions to the equation.

The discriminant is a specific part of the quadratic formula, denoted as \( b^2 - 4ac \), and plays a crucial role in determining the nature of the roots of a quadratic equation. The calculation outcome helps to predict whether the equation has two real and distinct roots, one repeated real root, or two complex roots.

Real and Distinct Roots

If the discriminant is positive, this indicates that the quadratic equation has two different real roots.

Repeated Real Root

If the discriminant is zero, the roots are real and identical, often referred to as a repeated or double root.

Complex Roots

When the discriminant is negative, as in our exercise \( (-15) \), it implies that the quadratic equation has two complex roots. Complex roots come in conjugate pairs. Understanding the discriminant offers a valuable forecast about the solutions before any actual computation is done.

Complex numbers extend the idea of the one-dimensional number line into the two-dimensional complex plane by including the square root of negative one, denoted as \( i \), which is called the imaginary unit. A complex number is generally expressed in the form \( a + bi \), with \( a \) being the real part and \( bi \) the imaginary part.

In the context of quadratic equations, when the discriminant is negative, the quadratic formula will result in the square root of a negative number. This is where complex numbers come into play, as they allow for the expression of these roots in a form that combines both real and imaginary numbers. In our exercise, since the discriminant \( -15 \) is negative, the solutions involve the square root of \( 15 \) multiplied by the imaginary unit \( i \) . This concept is fundamental in higher mathematics and physics, as it enables the resolution of problems that require solutions beyond the realm of real numbers.