Problem 109 - 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 of the quadratic equation are complex: \(x = \frac{5 \pm i\sqrt{47}}{6}\).

Step 1: Identify the Quadratic Equation

The given quadratic equation is \(3x^2 - 5x + 6 = 0\). Quadratic equations are in the form \(ax^2 + bx + c = 0\), where \(a = 3\), \(b = -5\), and \(c = 6\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Substitute \(a = 3\), \(b = -5\), and \(c = 6\) into this formula: \[(-5)^2 - 4(3)(6) = 25 - 72 = -47\]

Step 3: Determine the Nature of the Solutions

Since the discriminant \(-47\) is negative, the solutions of the quadratic equation are complex numbers.

Step 4: Use the Quadratic Formula

To solve the quadratic equation, use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Substituting the values \(a = 3\), \(b = -5\), and \(c = 6\): \[x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(6)}}{2(3)} = \frac{5 \pm \sqrt{-47}}{6}\]

Step 5: Express the Complex Solutions

Since \(\sqrt{-47}\) is a complex number, it can be written as \(i\sqrt{47}\), where \(i\) is the imaginary unit. Thus, the solutions can be expressed as: \[x = \frac{5 \pm i\sqrt{47}}{6}\]

Key Concepts

Discriminant

In quadratic equations, the discriminant plays a crucial role in determining the nature of the solutions. The discriminant is calculated using the formula \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

The value of the discriminant provides insight into the type of solutions:

• If the discriminant is positive, the equation has two distinct real solutions.
• If it is zero, there is exactly one real solution, or a repeated real root.
• If the discriminant is negative, the solutions are complex numbers.

In our example, the discriminant is negative (\(-47\)), indicating the solutions are not real but involve complex numbers.

Complex Numbers

Complex numbers expand the scope of numbers and allow for the solution of equations that do not have real solutions. A complex number is of the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. The unit \(i\) stands for \(\sqrt{-1}\).

When a quadratic equation has a negative discriminant, the square root of a negative number occurs, which is where complex numbers come in. In our exercise, the term \(\sqrt{-47}\) is rewritten using \(i\), making it \(i\sqrt{47}\). This is part of the solution pair indicating that the quadratic equation does not intersect the real number line.

Understanding complex numbers is crucial for higher mathematics, engineering, and physics as they often describe phenomena where two-dimensional components are essential.

Quadratic Formula

The quadratic formula is a one-stop solution to find the roots of any quadratic equation \(ax^2 + bx + c = 0\). This universally applicable formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute the values of \(a\), \(b\), and \(c\) directly into this formula to find the solutions.

The "\(\pm\)" sign signifies the two potential solutions. While using this formula in our exercise, replacing the expression under the square root \(b^2 - 4ac\), determines if we handle real or complex roots.

Since the \discriminant in our example is negative (\(-47\)), it results in complex solutions. Substituting into the formula, and simplifying, we obtain the results

• \(x = \frac{5 + i\sqrt{47}}{6}\)
• \(x = \frac{5 - i\sqrt{47}}{6}\)

The quadratic formula is powerful and a standard method, widely used across various mathematical applications and problem-solving scenarios.