Problem 14 - Elementary Algebra

Question

Calculate the discriminant and use it to determine the number and type of solutions. Do not solve. $$ 9 t 2-6 t+1=0 $$

Step-by-Step Solution

Answer

One real repeated solution.

Step 1: Identify coefficients

For the quadratic equation in the standard form \( ax^2 + bx + c = 0 \), identify the coefficients from the given quadratic equation \( 9t^2 - 6t + 1 = 0 \). Here, \( a = 9 \), \( b = -6 \), and \( c = 1 \).

Step 2: Calculate the discriminant

The discriminant \( \Delta \) of a quadratic equation is calculated using the formula \( \Delta = b^2 - 4ac \). Substitute the identified coefficients into the formula: \( \Delta = (-6)^2 - 4 \cdot 9 \cdot 1 \).

Step 3: Simplify the discriminant expression

First, calculate \( (-6)^2 = 36 \) and \( 4 \cdot 9 \cdot 1 = 36 \). Substitute these values back into the discriminant formula: \( \Delta = 36 - 36 = 0 \).

Step 4: Determine the number and type of solutions

Since the discriminant \( \Delta = 0 \), the quadratic equation has exactly one real and repeated solution (a double root).

Key Concepts

Understanding the Discriminant

In quadratic equations, the discriminant is a key mathematical expression used to determine the nature and the number of solutions. It is denoted by the symbol \( \Delta \) and is calculated using the formula \( \Delta = b^2 - 4ac \). This formula involves the coefficients \( a \), \( b \), and \( c \) of the quadratic equation in the standard form \( ax^2 + bx + c = 0 \). The discriminant can inform us about the solutions without actually solving the equation.

After computing \( \Delta \):

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution, known as a double root.
• If \( \Delta < 0 \), there are no real solutions, but two complex solutions.

In the provided example, \( \Delta = 0 \) indicates one real and repeated solution, reflecting the elegance of this mathematical concept in predicting outcomes.

Role and Identification of Coefficients

The coefficients in a quadratic equation are the numerical factors that multiply the variables. These coefficients are crucial as they form the basis for calculating the discriminant. A quadratic equation typically appears in the form \( ax^2 + bx + c = 0 \). Here:

• \( a \): the coefficient of \( x^2 \), controls the parabola's width and direction.
• \( b \): the coefficient of \( x \), affects the symmetry and roots of the parabola.
• \( c \): the constant term, shifts the parabola up or down.

In the exercise, the equation \( 9t^2 - 6t + 1 = 0 \) gives us: - \( a = 9 \) - \( b = -6 \) - \( c = 1 \) Understanding these coefficients helps in calculating the discriminant to predict the nature of the solutions easily.

Identifying Real Solutions

Real solutions of a quadratic equation correspond to the values of the variable that satisfy the equation. These solutions are points where the graph of the equation, a parabola, crosses the x-axis. The discriminant plays a crucial role in determining the number of real solutions a quadratic equation will have.
When the discriminant \( \Delta \) is:

• Greater than zero \((\Delta > 0)\): the quadratic equation has two distinct real solutions.
• Equal to zero \((\Delta = 0)\): there is one real solution, creating a double root scenario.
• Less than zero \((\Delta < 0)\): no real solutions exist, indicating complex roots.

In our example, \( \Delta = 0 \), thus, the equation has one real solution. This means the parabola touches the x-axis at just one point, demonstrating the fundamental insights provided by examining the discriminant.