Problem 15 - 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. $$3 y^{2}=3 y+2$$

Step-by-Step Solution

Answer

The solutions for \( y \) are \( y = \frac{3 + \sqrt{33}}{6} \) and \( y = \frac{3 - \sqrt{33}}{6} \).

Step 1: Rewriting Equation to Standard Form

Start by rewriting the given quadratic equation into the standard form, which is written as \( ax^2 + bx + c = 0 \). The given equation is \( 3y^2 = 3y + 2 \). Subtract \( 3y + 2 \) from both sides to get \( 3y^2 - 3y - 2 = 0 \).

Step 2: Identifying a, b, and c

Identify the coefficients \( a \), \( b \), and \( c \) from the standard form equation \( 3y^2 - 3y - 2 = 0 \). Compare it to \( ax^2 + bx + c = 0 \) to see that \( a = 3 \), \( b = -3 \), and \( c = -2 \).

Step 3: Using the Quadratic Formula

The quadratic formula is \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the values of \( a \), \( b \), and \( c \) into the formula: \( y = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 3 \cdot (-2)}}{2 \cdot 3} \).

Step 4: Simplifying the Expression

Simplify inside the square root: \( (-3)^2 = 9 \) and \( 4 \cdot 3 \cdot (-2) = -24 \). Thus, the expression inside the square root becomes \( 9 - (-24) = 9 + 24 = 33 \).

Step 5: Solving for y

The simplified quadratic formula is \( y = \frac{3 \pm \sqrt{33}}{6} \). Split this into the two possible solutions: \( y = \frac{3 + \sqrt{33}}{6} \) and \( y = \frac{3 - \sqrt{33}}{6} \). Both are simplified forms for \( y \).

Key Concepts

Quadratic Formula

When faced with a quadratic equation of the form \( ax^2 + bx + c = 0 \), the quadratic formula is a powerful tool to find its solutions. The formula is:

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

This formula provides the solutions or 'roots' of the quadratic equation.
The term under the square root, \( b^2 - 4ac \), is called the discriminant. It helps determine the nature of the roots:

• If the discriminant is positive, there are two distinct real solutions.
• If it equals zero, there is exactly one real solution.
• If it's negative, the solutions are complex or imaginary numbers.

Learning to use the quadratic formula effectively takes practice, but it is an invaluable tool in algebra.

Standard Form of a Quadratic Equation

Before applying the quadratic formula, it's essential to write the quadratic equation in its standard form: \( ax^2 + bx + c = 0 \). This form clearly presents the coefficients \( a \), \( b \), and \( c \), which are necessary to use the quadratic formula effectively.
Rewriting an equation to standard form involves moving all terms to one side of the equation, leaving zero on the other side.
For the given example, starting with \( 3y^2 = 3y + 2 \), you would subtract \( 3y + 2 \) from both sides: resulting in \( 3y^2 - 3y - 2 = 0 \).
This not only facilitates solving the equation but also enhances your understanding of how each component interacts in the equation.

Solving Equations

To solve a quadratic equation, especially using the quadratic formula, follow these steps:

• Rearrange the equation to standard form: \( ax^2 + bx + c = 0 \).
• Identify the coefficients \( a \), \( b \), and \( c \).* In our example, these are \( a = 3 \), \( b = -3 \), and \( c = -2 \).
• Substitute these values into the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Calculate the discriminant \( b^2 - 4ac \) to find the type and number of solutions.
• Simplify the expression, compute the final values, and write down the possible solutions.

Consistent practice with these steps prepares you to approach any quadratic equations confidently.

Mathematics Education

Understanding and using quadratic equations is a fundamental skill in mathematics education. It lays the groundwork for exploring more complex algebraic concepts. Mastery of these equations aids in developing logical thinking and problem-solving skills.
In classroom settings, students often practice quadratic equations through exercises that enhance understanding and encourage exploration.

• Solving quadratic equations improves critical thinking.
• It reinforces mathematical concepts like factoring and completing the square.
• Students gain confidence as they understand the application of quadratic equations in real-world problems.

Educators often use a variety of techniques, including visualization and application-based learning, to demystify quadratic equations, making them more accessible to learners across different skill levels.