Problem 99 - Calculus for Biology and Medicine

Question

Solve each quadratic equation in the complex number system. \(2 x^{2}-3 x+2=0\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3 + i \sqrt{7}}{4}\) and \(x = \frac{3 - i \sqrt{7}}{4}\).

Step 1: Identify the Coefficients

First, let's identify the coefficients of the quadratic equation given as \(2x^2 - 3x + 2 = 0\). The coefficients are \(a = 2\), \(b = -3\), and \(c = 2\).

Step 2: Use the Quadratic Formula

We will use the quadratic formula to solve for \(x\). The formula is \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). Substitute \(a = 2\), \(b = -3\), and \(c = 2\) into this formula.

Step 3: Calculate the Discriminant

The discriminant, \(b^2 - 4ac\), helps us determine the nature of the roots. Calculate it as follows: \((-3)^2 - 4(2)(2) = 9 - 16 = -7\). Since the discriminant is negative, the equation has complex roots.

Step 4: Compute the Square Root of the Discriminant

Because the discriminant is negative, we need the square root of \(-7\) which is \(\sqrt{-7} = i\sqrt{7}\). We use this in the quadratic formula.

Step 5: Substitute Into the Quadratic Formula and Solve

Now substitute the calculated values into the quadratic formula: \[ x = \frac{{3 \pm i \sqrt{7}}}{4} \]. This gives two solutions: \[ x = \frac{3 + i \sqrt{7}}{4} \] and \[ x = \frac{3 - i \sqrt{7}}{4} \].

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and they involve polynomial expressions that include a variable raised to the second power. Generally, a quadratic equation is written in the form \( ax^2 + bx + c = 0 \). Here, \(a\), \(b\), and \(c\) are constants where \(a eq 0\). Quadratics can have different types of roots based on the discriminant value:

• Two distinct real roots
• One repeated real root
• Two complex roots

In the complex number system, we often encounter complex roots when solving quadratic equations with a negative discriminant. This is due to the presence of the imaginary unit \(i\), which is defined as the square root of \(-1\). When graphing quadratic equations, the roots correspond to the points where the parabola intersects the x-axis. If the roots are complex, the parabola does not intersect the x-axis in the real plane. Understanding how to solve these equations forms the basis for more complex algebraic concepts.

Discriminant

The discriminant is a crucial component of a quadratic equation, as it informs us about the nature of the roots. It is derived from the quadratic formula and is defined as \(b^2 - 4ac\). Here’s what the discriminant reveals:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the equation results in two complex roots.

In our exercise, the calculated discriminant was \(-7\), indicating complex roots. This outcome tells us there is no real intersection with the x-axis, leading us into the realm of complex numbers to find the precise solutions. The discriminant not only demonstrates the root type but also hints at the geometric nature of the parabola associated with the quadratic equation.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]By substituting the coefficients \(a\), \(b\), and \(c\) from any specific equation into this formula, we can easily find the solutions for \(x\). The plus-minus (\(\pm\)) symbol indicates that there are typically two solutions for the equation.Using the quadratic formula provides not only a method for finding real roots but also allows us to handle complex roots efficiently. When the discriminant is negative, the quadratic formula accounts for this by including the imaginary unit \(i\), resulting in complex solutions.In the exercise provided, after substituting \(a = 2\), \(b = -3\), and \(c = 2\), and calculating the discriminant, we used the quadratic formula to find the complex roots:

• \( x = \frac{3 + i \sqrt{7}}{4} \)
• \( x = \frac{3 - i \sqrt{7}}{4} \)

The quadratic formula ensures that we have a reliable method to solve quadratic equations no matter the nature of their discriminant.