Problem 111 - Calculus for Biology and Medicine

Question

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(3 x^{2}-5 x+6=0\)

Step-by-Step Solution

Answer

The solutions are complex: \(x = \frac{5 \pm \sqrt{47}i}{6}\).

Step 1: Identify the Discriminant

The discriminant of a quadratic equation in the form \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Here, \(a = 3\), \(b = -5\), and \(c = 6\). Calculate the discriminant: \((-5)^2 - 4 \cdot 3 \cdot 6\).

Step 2: Calculate the Discriminant

Compute the value of the discriminant: \(25 - 72 = -47\).

Step 3: Determine the Nature of the Solutions

Since the discriminant is negative (-47), the quadratic equation has two complex conjugate solutions.

Step 4: Apply the Quadratic Formula

Use the quadratic formula to find the roots of the equation: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting the values, we get \(x = \frac{-(-5) \pm \sqrt{-47}}{6}\).

Step 5: Simplify the Expression

Simplify the expression further: \(x = \frac{5 \pm \sqrt{47}i}{6}\), where \(i\) is the imaginary unit.

Key Concepts

Discriminant

The discriminant is a crucial part of the quadratic equation. It's like a teller that reveals the nature of the solutions you'll find. For any quadratic equation in the form \(ax^2 + bx + c = 0\), the discriminant is calculated as \(b^2 - 4ac\). The values of \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation. Here's what the discriminant tells you:

• If it's positive, you have two distinct real solutions.
• If it's zero, there’s exactly one real solution, which is repeated.
• If it's negative, like in our exercise where it is \(-47\), the solutions are complex.

In the context of our given equation \(3x^2-5x+6=0\), identifying the discriminant as negative means knowing in advance that the solutions aren't real numbers. This helps us anticipate the type of numbers we will be dealing with before even solving the equation.

Complex Solutions

Complex solutions might sound tricky, but they are just as manageable as real numbers. They pop up in quadratic equations when the discriminant is negative, indicating that no real number solves the equation. Complex numbers take the form \(a + bi\) where \(i\) is the imaginary unit, satisfying \(i^2 = -1\).When encountering a negative discriminant, as we did with \(-47\), it means the square root of a negative number needs handling. Using \(i\) helps here. For example, \(\sqrt{-47} = \sqrt{47}i\). The solutions emerge from placing these within the larger equation, resulting in two complex conjugates. In our exercise, we found them as \(\frac{5 + \sqrt{47}i}{6}\) and \(\frac{5 - \sqrt{47}i}{6}\). These are mirror images across the real axis, making them complex conjugates.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It’s given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula works with any quadratic equation because it utilizes the coefficients \(a\), \(b\), and \(c\). The symbol \(\pm\) hints at two solutions—one using the plus sign and the other using the minus.Before diving into the solving, the discriminant part \(b^2 - 4ac\) should be calculated first, as seen in our steps. If it’s negative, the solutions will involve complex numbers. In our exercise, substituting the values yields:\[x = \frac{-(-5) \pm \sqrt{-47}}{6} = \frac{5 \pm \sqrt{47}i}{6}\]This reveals the two complex solutions we derived. By applying this formula methodically, you maintain control over the process of solving any quadratic equation.