Problem 20 - College Algebra with Modeling and Visualization

Question

Exercises \(1-28:\) Solve the quadratic equation. Check your answers for Exercises \(1-12\). $$ \frac{3}{4} x^{2}+\frac{1}{2} x-\frac{1}{2}=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{-1 + \sqrt{13}}{3} \) and \( x = \frac{-1 - \sqrt{13}}{3} \).

Step 1: Identify the coefficients

First, identify the coefficients from the given quadratic equation, which is \[ \frac{3}{4}x^{2} + \frac{1}{2}x - \frac{1}{2} = 0 \]The quadratic equation format is \( ax^2 + bx + c = 0 \). Thus:\[ a = \frac{3}{4}, \quad b = \frac{1}{2}, \quad c = -\frac{1}{2} \]

Step 2: Calculate the discriminant

Calculate the discriminant using the formula:\[ D = b^2 - 4ac \]Substitute the values found:\[ D = \left(\frac{1}{2}\right)^2 - 4\left(\frac{3}{4}\right)\left(-\frac{1}{2}\right) \]Simplify the expression:\[ D = \frac{1}{4} + 3 = \frac{1}{4} + \frac{12}{4} = \frac{13}{4} \]

Step 3: Apply the quadratic formula

Use the quadratic formula to find the roots:\[ x = \frac{-b \pm \sqrt{D}}{2a} \]Substitute the known values of \( b \), \( D \), and \( a \):\[ x = \frac{-\frac{1}{2} \pm \sqrt{\frac{13}{4}}}{2 \times \frac{3}{4}} \]Simplify:\[ x = \frac{-\frac{1}{2} \pm \frac{\sqrt{13}}{2}}{\frac{3}{2}} \]Multiply numerator and denominator by 2:\[ x = \frac{-1 \pm \sqrt{13}}{3} \]

Step 4: State the solutions

From the quadratic formula, the solutions are:\[ x_1 = \frac{-1 + \sqrt{13}}{3} \]\[ x_2 = \frac{-1 - \sqrt{13}}{3} \]

Step 5: Verify the solutions

Substitute the solutions back into the original equation to verify.For \( x_1 \):Replace \( x \) with \( \frac{-1 + \sqrt{13}}{3} \) in the original equation and simplify to ensure it satisfies the equation.For \( x_2 \):Replace \( x \) with \( \frac{-1 - \sqrt{13}}{3} \) in the original equation and simplify to ensure it satisfies the equation.Both should equal zero, confirming the solutions.

Key Concepts

Discriminant

In the realm of quadratic equations, the discriminant plays a crucial role. It helps determine the nature of the roots without actually solving the equation.
The discriminant (denoted as \( D \)) is calculated using the formula:

• \( D = b^2 - 4ac \)

Here, \( a \), \( b \), and \( c \) are the polynomial coefficients from the quadratic equation of the form \( ax^2 + bx + c = 0 \).
The discriminant gives insights into the type of roots:

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has one real root, often called a repeated or double root.
• If \( D < 0 \), the equation has two complex roots.

In the given exercise, substituting the values \( b = 1/2 \), \( a = 3/4 \), and \( c = -1/2 \) into the formula gives \( D = 13/4 \). Since \( D > 0 \), the equation has two real roots.

Quadratic Formula

The quadratic formula is a straightforward and reliable method to find the roots of any quadratic equation.
This powerful formula is:

• \( x = \frac{-b \pm \sqrt{D}}{2a} \)

Here, \( D \) stands for the discriminant (\( b^2 - 4ac \)), and \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation.
When solving a quadratic equation:1. First calculate the discriminant \( D \).2. Substitute \( b \), \( a \), and the calculated \( \sqrt{D} \) into the quadratic formula.
For the equation \( \frac{3}{4}x^2 + \frac{1}{2}x - \frac{1}{2} = 0 \),

• \( b = 1/2 \)
• \( a = 3/4 \)
• \( D = 13/4 \)
• Plugging these into the formula gives the two solutions:
• \( x_1 = \frac{-1 + \sqrt{13}}{3} \)
• \( x_2 = \frac{-1 - \sqrt{13}}{3} \)

Polynomial Coefficients

Polynomial coefficients are the numbers \( a \), \( b \), and \( c \) in a quadratic equation \( ax^2 + bx + c = 0 \).
These coefficients play a significant part in controlling the shape and position of a parabola, which is the graph of a quadratic equation.

• \( a \) is the quadratic coefficient and affects the direction and width of the parabola:
• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.
• Larger \( |a| \) values result in a narrower parabola.
• \( b \) is the linear coefficient, influencing the tilt or the axis intercepts of the parabola.
• \( c \) is the constant term, representing the y-intercept where the graph intersects the y-axis.

In the original exercise \( \frac{3}{4}x^2 + \frac{1}{2}x - \frac{1}{2} = 0 \), the coefficients are:

• \( a = \frac{3}{4} \)
• \( b = \frac{1}{2} \)
• \( c = -\frac{1}{2} \)

Understanding these coefficients provides a deeper insight into solving quadratic equations and predicting the parabolic graph's properties.