Problem 11 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$8 s^{2}+20 s=12$$

Step-by-Step Solution

Answer

The roots are \( s = \frac{1}{2} \) and \( s = -3 \).

Step 1: Write the quadratic in standard form

First, we need to rewrite the equation in the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). For the given equation \( 8s^2 + 20s = 12 \), subtract 12 from both sides to get: \( 8s^2 + 20s - 12 = 0 \).

Step 2: Identify coefficients a, b, c

From the standard form equation \( 8s^2 + 20s - 12 = 0 \), identify the coefficients: \( a = 8 \), \( b = 20 \), and \( c = -12 \).

Step 3: Recall the quadratic formula

The quadratic formula is used to find the roots of a quadratic equation and is given by: \( s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 4: Calculate the discriminant

The discriminant is the expression under the square root in the quadratic formula: \( b^2 - 4ac \). Substitute the values of \( a \), \( b \), and \( c \) to find the discriminant: \( 20^2 - 4 \times 8 \times (-12) \). This calculates to \( 400 + 384 = 784 \).

Step 5: Calculate the square root of the discriminant

Since the discriminant is \( 784 \), calculate its square root: \( \sqrt{784} = 28 \).

Step 6: Apply the quadratic formula

Substitute \( b = 20 \), the square root of the discriminant (\( \sqrt{784} = 28 \)), and \( a = 8 \) into the quadratic formula: \[ s = \frac{-20 \pm 28}{16} \].

Step 7: Determine the roots

Compute the two possible values of \( s \): \( s_1 = \frac{-20 + 28}{16} = \frac{8}{16} = \frac{1}{2} \) and \( s_2 = \frac{-20 - 28}{16} = \frac{-48}{16} = -3 \).

Key Concepts

Quadratic Formula

To solve quadratic equations, one powerful tool at our disposal is the quadratic formula. This formula is a reliable method to find the roots of any quadratic equation, provided it is in the standard form. The quadratic formula is given by the expression:
\[ s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where:

• \(a\), \(b\), and \(c\) are coefficients in the quadratic equation,
• \(s\) are the unknown variable solutions or roots we wish to find.

The plus-minus symbol (\(\pm\)) indicates there may be two possible solutions for \(s\). These are called the roots of the equation. This is what makes quadratic equations interesting and versatile: they often have two solutions.

Standard Form

The standard form of a quadratic equation is crucial to correctly applying the quadratic formula. It is expressed as:
\[ ax^2 + bx + c = 0 \]
This form lays out the equation in terms of three coefficients: \(a\), \(b\), and \(c\). To convert any quadratic equation to the standard form, make sure all terms are on one side of the equation, leaving zero on the other side. This is exactly what we did in the given problem:

• The original equation was \(8s^2 + 20s = 12\).
• We subtracted \(12\) from both sides to yield \(8s^2 + 20s - 12 = 0\).

Remember, the equation must equal zero for the quadratic formula to work. This ensures the equation is balanced and highlights the relationship between all terms.

Discriminant

The discriminant is an integral part of the quadratic formula, found under the square root. It is denoted by:
\[ b^2 - 4ac \]
The discriminant gives us important information about the nature of the roots:

• If the discriminant is positive, \(b^2 - 4ac > 0\), there are two distinct real roots.
• If the discriminant is zero, \(b^2 - 4ac = 0\), there is exactly one real root (a repeated root).
• If the discriminant is negative, \(b^2 - 4ac < 0\), there are no real roots, but two complex roots instead.

In our example, the discriminant calculated to \(784\), which is positive. This means we have two distinct real roots, as verified by the solutions \(s = \frac{1}{2}\) and \(s = -3\).

Roots of a Quadratic

The term "roots" refers to the solutions of the quadratic equation, where the full expression equals zero. Finding these roots tells us where the graph of the quadratic equation intersects the x-axis.
In the example we solved, after applying the quadratic formula, we found two values for \(s\):

• \(s_1 = \frac{1}{2}\)
• \(s_2 = -3\)

These roots indicate the points where the quadratic function crosses or touches the x-axis.
Knowing the roots provides insights into the function's graphical behavior. It also helps in identifying possible maximum and minimum values, as well as the direction in which the graph opens (upward or downward). This understanding is essential for analyzing quadratic functions in various scenarios.