Problem 24 - 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. $$20 r^{2}=20 r+1$$

Step-by-Step Solution

Answer

The solutions are \( r = \frac{1}{2} + \frac{\sqrt{30}}{10} \) and \( r = \frac{1}{2} - \frac{\sqrt{30}}{10} \).

Step 1: Rewrite the Equation in Standard Form

The first step in solving a quadratic equation using the quadratic formula is to ensure the equation is in standard form, which is \( ax^2 + bx + c = 0 \). Begin by subtracting \( 20r + 1 \) from both sides: \[ 20r^2 - 20r - 1 = 0 \] Thus, our equation is now in the form \( ax^2 + bx + c = 0 \) with \( a = 20 \), \( b = -20 \), and \( c = -1 \).

Step 2: Identify Coefficients

Now that the equation is in the form \( ax^2 + bx + c = 0 \), identify the coefficients: \( a = 20 \), \( b = -20 \), and \( c = -1 \). These values will be used in the quadratic formula.

Step 3: Apply the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the coefficients into the formula: \[ r = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 20 \cdot (-1)}}{2 \cdot 20} \] This simplifies to: \[ r = \frac{20 \pm \sqrt{400 + 80}}{40} \] \[ r = \frac{20 \pm \sqrt{480}}{40} \]

Step 4: Simplify Under the Square Root

Calculate the expression under the square root, \( \sqrt{480} \). First, simplify \( \sqrt{480} \) as follows: \[ \sqrt{480} = \sqrt{16 \cdot 30} = 4\sqrt{30} \] Thus, the equation becomes: \[ r = \frac{20 \pm 4\sqrt{30}}{40} \]

Step 5: Simplify the Expression

We can further simplify the fraction: \[ r = \frac{20}{40} \pm \frac{4\sqrt{30}}{40} \] \[ r = \frac{1}{2} \pm \frac{\sqrt{30}}{10} \] Therefore, the solutions are: \[ r = \frac{1}{2} + \frac{\sqrt{30}}{10} \text{ and } r = \frac{1}{2} - \frac{\sqrt{30}}{10} \]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations. A quadratic equation generally looks like this: \( ax^2 + bx + c = 0 \). To find the roots, or solutions, for the variable \( x \), we use the quadratic formula:

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

This formula provides two possible solutions because of the "\( \pm \)" symbol, which means you calculate once with a plus and once with a minus. There are three possible outcomes for the roots:

• Two real and distinct roots.
• One real, repeated root.
• No real roots, which means the roots are complex numbers.

In our problem, the quadratic formula helps us determine the values of \( r \) that satisfy the equation, once transformed.

Standard Form

The standard form of a quadratic equation is crucial for using the quadratic formula. It requires the equation to be neatly written as \( ax^2 + bx + c = 0 \). Here:

• \( a \), \( b \), and \( c \) are constants known as coefficients.
• \( x \) stands for the variable to solve.

In our example, the original equation was \( 20r^2 = 20r + 1 \). Before applying the quadratic formula, it was transformed into: \( 20r^2 - 20r - 1 = 0 \). This transformation involved moving all terms to one side of the equation and equating it to zero.

Coefficients

In quadratic equations, identifying coefficients is an important step. Coefficients are the numbers in front of the variable terms. In the standard form \( ax^2 + bx + c = 0 \):

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

For the equation \( 20r^2 - 20r - 1 = 0 \), the coefficients are:

• \( a = 20 \)
• \( b = -20 \)
• \( c = -1 \)

Identifying these coefficients correctly is essential, as they are the ingredients that go into the quadratic formula.

Simplifying Radicals

Simplifying radicals is key to obtaining the final and most concise solution to a quadratic equation. A radical is a term under the square root sign \( \sqrt{} \). In our case, we initially had \( \sqrt{480} \).To simplify a radical like \( \sqrt{480} \), find the largest square number that divides it. For \( 480 \), this is \( 16 \):

• \( \sqrt{480} = \sqrt{16 \times 30} = \sqrt{16} \times \sqrt{30} = 4\sqrt{30} \)

This simplification reduced the expression under the square root into a simpler form. Once simplification is complete, you then substitute back into the formula to find simplified values for \( x \). In this problem, this simplification step ensured the final solutions were as clear as possible.