Problem 21 - Basic Technical Mathematics with Calculus

Question

solve the given quadratic equations, using the quadratic formula. Exercises \(5-8\) are the same as Exercises \(11-14\) of Section 7.2. $$8 t^{2}+61 t=-120$$

Step-by-Step Solution

Answer

The equation has no real roots; it has two complex roots.

Step 1: Understanding the Given Equation

The given quadratic equation is in the form of \(8t^2 + 61t = -120\). This equation is not in the standard form \(ax^2 + bx + c = 0\). We need to rearrange it first.

Step 2: Rearrange to Standard Form

To convert the equation into the standard form, move all terms to one side of the equation, resulting in \(8t^2 + 61t + 120 = 0\).

Step 3: Identify Coefficients

From the standard form \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 8\), \(b = 61\), and \(c = 120\).

Step 4: Apply the Quadratic Formula

The quadratic formula is \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values of \(a\), \(b\), and \(c\): \(t = \frac{-61 \pm \sqrt{61^2 - 4 \times 8 \times 120}}{2 \times 8}\).

Step 5: Calculate the Discriminant

Calculate the discriminant using \(b^2 - 4ac\): \(61^2 - 4 \times 8 \times 120 = 3721 - 3840 = -119\).

Step 6: Determine Nature of Roots

The discriminant \(-119\) is negative, which means the equation has no real roots but two complex roots.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to solve quadratic equations. A quadratic equation is typically written in the standard form, which is \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients, where \(a\) cannot be zero. The quadratic formula allows us to find the values of \(x\) (or in our example, \(t\)) that satisfy the equation.

• The formula itself is: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• The symbol \(\pm\) means that the formula gives two solutions: one by adding the square root part and one by subtracting it.

Using this formula is particularly useful because it provides a direct way to find the roots without needing to factor the quadratic expression. This is especially helpful when the equation is difficult or impossible to factor by inspection.

Standard Form Transformation

Before using the quadratic formula, a quadratic equation must be in standard form. This form is expressed as \(ax^2 + bx + c = 0\). If the equation is not initially in this form, you must rearrange it by moving all terms to one side until the equation is set to zero.
In our example, the original equation was \(8t^2 + 61t = -120\). To transform it into the standard form, you need to move \(-120\) to the left side, resulting in \(8t^2 + 61t + 120 = 0\). This step is essential as it allows us to correctly identify the coefficients \(a\), \(b\), and \(c\) needed for applying the quadratic formula.

Discriminant Calculation

The discriminant is a part of the quadratic formula that determines the nature of the roots of the quadratic equation. It is found using the expression \(b^2 - 4ac\). The discriminant tells us:

• If it is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root, sometimes called a double root.
• If it is negative, the equation has no real roots but instead has two complex roots.

For the equation \(8t^2 + 61t + 120 = 0\), the discriminant is calculated as \(61^2 - 4 \times 8 \times 120\). This simplifies to \(3721 - 3840 = -119\). Since the discriminant is negative, it indicates that the quadratic equation has complex roots.

Complex Roots

When a quadratic equation's discriminant is negative, the equation does not have real roots. Instead, it has complex roots. Complex numbers include a real part and an imaginary part and are usually expressed in the form \(a + bi\), where \(i\) is the square root of \(-1\).
In the context of the quadratic formula, having a negative discriminant involves taking the square root of a negative number, which leads to the imaginary component of the roots.

• Even though complex roots might seem abstract, they are a valid and important solution in mathematics.
• Complex roots always appear in conjugate pairs, meaning if one root is \(a + bi\), the other will be \(a - bi\).

For our example, the quadratic equation provides two complex solutions stemming from the negative discriminant calculated earlier.