Problem 37 - Algebra 2

Question

Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. \(\frac{3}{4} x^{2}-\frac{1}{3} x-1=0\)

Step-by-Step Solution

Answer

Discriminant: \( \frac{28}{9} \), two distinct real roots. Solutions: \( x = \frac{2(1 \pm \sqrt{28})}{9} \).

Step 1: Identify the Coefficients

To solve the quadratic equation \( \frac{3}{4} x^2 - \frac{1}{3} x - 1 = 0 \), identify the coefficients: \( a = \frac{3}{4} \), \( b = -\frac{1}{3} \), and \( c = -1 \).

Step 2: Calculate the Discriminant

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). Substitute the values: \[ \Delta = \left(-\frac{1}{3}\right)^2 - 4 \left(\frac{3}{4}\right)(-1) \]. Simplify to find \( \Delta = \frac{1}{9} + 3 = \frac{28}{9} \).

Step 3: Determine the Number and Type of Roots

The discriminant \( \Delta = \frac{28}{9} \) is greater than 0, indicating that the quadratic equation has two distinct real roots.

Step 4: Apply the Quadratic Formula

The solutions to the quadratic equation are given by the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]. Substitute the known values: \( b = -\frac{1}{3} \), \( a = \frac{3}{4} \), and \( \Delta = \frac{28}{9} \), resulting in: \[ x = \frac{\frac{1}{3} \pm \sqrt{\frac{28}{9}}}{\frac{3}{2}} \].

Step 5: Simplify the Solutions

First, calculate \( \sqrt{\frac{28}{9}} = \frac{\sqrt{28}}{3} \). Then, substitute back to get: \[ x = \frac{\frac{1}{3} + \frac{\sqrt{28}}{3}}{\frac{3}{2}} \] and \[ x = \frac{\frac{1}{3} - \frac{\sqrt{28}}{3}}{\frac{3}{2}} \]. Simplify each expression: \[ x_1 = \frac{2(1 + \sqrt{28})}{9} \] and \[ x_2 = \frac{2(1 - \sqrt{28})}{9} \].

Key Concepts

Discriminant

The discriminant is a key component in solving quadratic equations. It helps determine the nature of the roots of the equation. In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(\Delta\) is calculated using the formula \(\Delta = b^2 - 4ac\). To compute it, we substitute the coefficients \(a\), \(b\), and \(c\) into this formula.

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root (a repeated root).
• If \(\Delta < 0\), the roots are complex and not real.

In our problem, the discriminant is \(\Delta = \frac{28}{9}\), which is greater than zero. This indicates the equation has two distinct real roots.

Real Roots

Understanding real roots is crucial when solving quadratic equations. Real roots are values of \(x\) that satisfy the equation, making it equal to zero.Here are the scenarios for an equation based on the discriminant:

• When \(\Delta > 0\), the quadratic has two different real roots, as seen in our example.
• When \(\Delta = 0\), there is one real root, forming a perfect square.
• When \(\Delta < 0\), the solution comprises complex roots, which do not intersect the x-axis.

In the given equation with \(\Delta = \frac{28}{9}\), the presence of this positive discriminant confirms the existence of two distinct real solutions.

Quadratic Formula

The quadratic formula is a universal method for finding solutions to quadratic equations. When in doubt, you can apply this formula to any quadratic equation of the form \(ax^2 + bx + c = 0\).The formula is expressed as:\[x = \frac{-b \pm \sqrt{\Delta}}{2a}\]This formula can find the roots once you know the discriminant \(\Delta\) and the coefficients \(a\), \(b\), and \(c\).Using our identified coefficients: \(a = \frac{3}{4}\), \(b = -\frac{1}{3}\), and our calculated discriminant \(\Delta = \frac{28}{9}\), we applied and solved the quadratic formula to get the solutions in the exercise example.

Coefficients

Coefficients in a quadratic equation are the numerical parameters associated with each term when expressed in the form \(ax^2 + bx + c = 0\). Identifying these coefficients correctly is critical to substitute into the discriminant and quadratic formula.

• \(a\): The coefficient of \(x^2\), which determines the parabola's openness direction.
• \(b\): The coefficient of \(x\), impacting the symmetry and position of the parabola.
• \(c\): The constant term, indicating the parabola's y-intercept on the graph.

For instance, the given equation \(\frac{3}{4} x^2 - \frac{1}{3} x - 1 = 0\) has \(a = \frac{3}{4}\), \(b = -\frac{1}{3}\), and \(c = -1\), playing crucial roles in further calculations.