Problem 66 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation. $$5 s^{2}+6 s+3=0$$

Step-by-Step Solution

Answer

The roots of the quadratic equation \(5s^2 + 6s + 3 = 0\) are complex and given by \( -0.6 \pm 1.55i \)

Step 1: Identify a, b, c

To start with, you must find out the values of a, b and c from the given quadratic equation. Here, a=5, b=6 and c=3, where a is the coefficient of \(s^2\), b is the coefficient of s and c is the constant.

Step 2: Calculate Discriminant

Next, you need to calculate the discriminant of the quadratic equation using the formula \(b^2 - 4ac\). Here, discriminant equals \(6^2 - 4*5*3 = 36 - 60 = -24\). Since the discriminant is negative, it means the roots of the equation are imaginary.

Step 3: Calculate the Roots

Afterwards, proceed by using the quadratic formula \(x = {-b \pm \sqrt{b^2 - 4ac}} / 2a \) to find the roots. Here \(s = {-6 \pm \sqrt{-24}} / 2*5 = -6/10 \pm \sqrt{-24}/10 = -0.6 \pm \sqrt{-2.4}\). It is evident that the roots are complex and can be written as \( -0.6 \pm 1.55i \).

Key Concepts

Discriminant

The discriminant is a concept that plays a crucial role in quadratic equations. It is a part of the quadratic formula that helps us determine the nature of the roots of a quadratic equation without actually solving it. The formula for the discriminant is given by: \( b^2 - 4ac \)
In this formula:

• b is the coefficient of the linear term
• a is the coefficient of the squared term
• c is the constant term

The value of the discriminant reveals a lot about the quadratic equation:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is precisely one real root (also known as a double root).
• When the discriminant is negative, like in our example where it is \(-24\), the roots of the quadratic equation are not real numbers but complex numbers.

Complex Roots

When the discriminant of a quadratic equation is negative, the roots are complex. Complex roots mean that the solution involves imaginary numbers. Let's explore what this means:
Complex numbers are numbers that have both a real part and an imaginary part. The imaginary unit is denoted as \(i\), and it is defined by the property \(i^2 = -1\). Thus, an imaginary number can be expressed in the form of \(bi\), where \(b\) is a real number.
In the context of quadratic equations, if you have a negative discriminant, the roots are of the form:

• \( -\frac{b}{2a} \pm \frac{\sqrt{-D}}{2a} \), where \(D\) is the discriminant.

For example, if the discriminant is \(-24\), the complex roots might look something like \(-0.6 \pm 1.55i\). Here, \(-0.6\) represents the real part of the complex number, and \(1.55i\) is the imaginary part. It’s important to note that complex roots always come in conjugate pairs for real quadratic equations, meaning if one root is \( a + bi\), the other will be \( a - bi\).

Quadratic Formula

The quadratic formula is a powerful tool to find the solutions of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula provides the roots based on the values of \(a\), \(b\), and \(c\) from the equation. Here’s a step-by-step guide on how it handles different discriminant cases:

• If the discriminant is positive, the formula will yield two distinct real solutions, using the plus-minus (\(\pm\)) operation to generate two different numbers.
• When it is zero, both operations result in the same number, giving a single real solution.
• If negative, as mentioned, the roots will be complex numbers. The presence of \(\sqrt{negative \: number}\) introduces the imaginary unit \(i\).

Therefore, the quadratic formula is not only a means to find answers but also to interpret and understand the nature of the roots. For our specific example, the formula reveals roots of the form \(-0.6 \pm 1.55i\), confirming the complex nature of the solution due to the negative discriminant.