Question

Describe the number and type of solutions when the value of the discriminant is negative.

Short answer

When the discriminant is negative, the quadratic equation has two complex (or imaginary) solutions.

Step-by-step solution

Understand the concept

The discriminant is the term \(b^2 - 4ac\) in the quadratic formula \(-b \pm \sqrt{{b^2 - 4ac}}\) / \(2a\). This is the term that is under the square root of this formula. The value of the discriminant can tell us the number and type of solutions. The discriminant can either be positive, zero or negative, and each case gives different types of solutions to the quadratic equation.

Determine the number and type of solutions when the discriminant is negative

When the discriminant is negative i.e. \(b^2 - 4ac < 0\), the quadratic equation has two complex solutions. It's because the square root of a negative number is an imaginary number. Therefore, any time the discriminant is negative, the solutions are complex or imaginary.

Complex solutions

Complex solutions are comprised of a real and an imaginary part and they are conjugate pairs. In this case, solutions are in the form \(x = a \pm bi\) where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with the property \(i^2 = -1\).

Key concepts

Complex Solutions

When delving into quadratic equations, understanding how the discriminant influences the solutions is crucial. A negative discriminant, realized when you calculate the term \(b^2 - 4ac\) and get a value less than zero, indicates that you're entering the realm of complex solutions. Unlike real numbers, complex solutions have two parts: a real part and an imaginary part, both of which are fundamental for expressing solutions that can't be represented on the traditional number line.

Imagine encountering a quadratic equation where no real number solutions seem to fit. In such scenarios, the solutions take the form \(x = a \pm bi\), where \(a\) represents the real part and \(bi\) represents the imaginary part, with \(i\) being the imaginary unit. It's noteworthy that these complex solutions always come in pairs, known as conjugate pairs, meaning if \(a + bi\) is a solution, then its counterpart \(a - bi\) will be the other solution. This ensures that the solutions are balanced with respect to both the real and imaginary components. So, when you stumble upon a negative discriminant, visualize a symphony of real and imaginary numbers teaming up to provide you with two intricate yet harmonious solutions.

Quadratic Formula

The quadratic formula, \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\), is like a key that unlocks every quadratic equation's secrets—it tells you the solutions, whether they're real numbers or complex numbers. It's designed to handle any quadratic equation of the form \(ax^2 + bx + c = 0\), smartly encompassing all the information you need about the solutions.

What's particularly intriguing about the quadratic formula is how it incorporates the discriminant \(\Delta = b^2 - 4ac\). The discriminant serves as a signal light—green for two distinct real solutions, yellow for one repeated real solution when \(\Delta = 0\), and red, signaling to proceed with caution, as you'll be entering the territory of complex solutions when \(\Delta < 0\). The beauty of the quadratic formula lies in its universality and its ability to systematically lead you to the solutions, guiding your journey through the problem with analytical clarity and precision.

Imaginary Numbers

Imaginary numbers may sound like a concept from a fantasy novel, but in mathematics, they are as real as it gets when it comes to solving certain equations. An imaginary number is defined as a number that gives a negative result when squared. The most fundamental imaginary number is \(i\), where \(i^2 = -1\). While it seems counterintuitive, this concept opens up a whole new dimension in mathematics, allowing the exploration of numbers beyond the restraints of the real number line.

In the context of quadratic equations, imaginary numbers come into play when you see a negative discriminant. Here, \(i\) is not just a fantastical idea but a practical tool to express square roots of negative numbers. For example, \(\sqrt{{-4}}\) would be expressed as \(2i\). These numbers are the building blocks of complex numbers, and they ensure that every quadratic equation has a solution, even if that solution doesn't exist on the traditional number axis. Indeed, imaginary numbers are a testament to the depth and flexibility of mathematical thought.