Problem 22 - 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. $$20 r^{2}=20 r+1$$

Step-by-Step Solution

Answer

The real roots are approximately \( r_1 = 1.04772 \) and \( r_2 = -0.04772 \).

Step 1: Write the equation in standard form

To apply the quadratic formula, we must first express the equation in standard quadratic form, which is \( ax^2 + bx + c = 0 \). For the equation \( 20r^2 = 20r + 1 \), subtract \( 20r \) and \( 1 \) from both sides to get \( 20r^2 - 20r - 1 = 0 \).

Step 2: Identify coefficients

From the standard form equation \( 20r^2 - 20r - 1 = 0 \), identify the coefficients: \( a = 20 \), \( b = -20 \), \( c = -1 \).

Step 3: Use the quadratic formula

The quadratic formula is \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the coefficients into the formula: \( r = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 20 \cdot (-1)}}{2 \cdot 20} \).

Step 4: Calculate the discriminant

Calculate the value of the discriminant \( b^2 - 4ac \): \((-20)^2 - 4 \times 20 \times (-1) = 400 + 80 = 480\).

Step 5: Compute the roots

With the discriminant calculated, find the roots: \( r = \frac{20 \pm \sqrt{480}}{40} \).Calculate \( \sqrt{480} \approx 21.9089 \), so the roots are:\( r_1 = \frac{20 + 21.9089}{40} \approx 1.04772 \)\( r_2 = \frac{20 - 21.9089}{40} \approx -0.04772 \).

Step 6: Verify the presence of real roots

The discriminant \( 480 \) is positive, indicating that there are two distinct real roots.

Key Concepts

Quadratic Formula

The quadratic formula is a critical tool for finding solutions to quadratic equations of the form \(ax^2 + bx + c = 0\). It provides an efficient way to discover whether the equation has real roots and what those roots are, without directly factoring the expression. The quadratic formula is expressed as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here’s a simpler way to think about it:

• \(-b\) is the opposite of the coefficient of the linear term \(x\).
• \(\pm\) indicates there are generally two possible solutions due to the quadratic nature.
• \(\sqrt{b^2 - 4ac}\) is called the discriminant, which determines the nature (real or complex) and number of roots.
• The denominator \(2a\) effectively scales the result to correspond with variable changes.

This single formula can solve any quadratic equation by just inserting the right coefficients \(a\), \(b\), and \(c\).

Discriminant

The discriminant plays a crucial role in determining the type of roots a quadratic equation will have. Calculated as \(b^2 - 4ac\), it helps evaluate:

• Whether roots are real or complex.
• If real, whether they are distinct or repeated.

The nature of the discriminant:

• A positive discriminant \((>0)\) indicates two distinct real roots.
• A zero discriminant \((=0)\) suggests exactly one real root, sometimes called a repeated root.
• A negative discriminant \((<0)\) implies that the roots are complex and not real.

In our example, the discriminant value is found to be 480, which is positive, indicating the equation has two distinct real roots.

Real Roots

Real roots are solutions to the quadratic equation that can be placed on a number line. They are typical outcomes of a quadratic expression when the discriminant is zero or positive. In simpler terms:

• When the discriminant \((b^2 - 4ac)\) is positive, it points to two distinct points on the line, meaning two different real numbers satisfy the equation.
• If it’s zero, both roots collapse into one, indicating a single solution where the parabola touches the x-axis.

In our solved equation, due to the positive discriminant of 480, two real roots exist:

• \(r_1 \approx 1.04772\)
• \(r_2 \approx -0.04772\)

These outcomes blend with other mathematical analysis to help visualize or solve real-world problems.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is a neat mathematical arrangement: \(ax^2 + bx + c = 0\). It is the conventional structure required before applying the quadratic formula effectively. Here’s why it’s important:

• Having it in this form aligns individual terms for easy identification of coefficients \(a\), \(b\), and \(c\).
• The standard form establishes a consistent method for recognizing and solving quadratics.
• It simplifies comparisons between different quadratic equations by unifying them under a common form.

For our case, the equation \(20r^2 = 20r + 1\) was reorganized to \(20r^2 - 20r - 1 = 0\), making it ready for solution attempts with known strategies like the quadratic formula.