Problem 66 - Algebra and Trigonometry

Question

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. \(9 x^{2}-4 x+\frac{4}{9}=0\)

Step-by-Step Solution

Answer

The equation has exactly one real solution.

Step 1: Identify the coefficients

For the quadratic equation in the form of \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 9\), \(b = -4\), and \(c = \frac{4}{9}\).

Step 2: Calculate the discriminant

The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Substitute \(b = -4\), \(a = 9\), and \(c = \frac{4}{9}\) to get \(D = (-4)^2 - 4 \times 9 \times \frac{4}{9}\).

Step 3: Simplify the discriminant

Calculate \((-4)^2 = 16\). Then calculate \(4 \times 9 = 36\), so \(36 \times \frac{4}{9} = 16\). Therefore, the discriminant \(D = 16 - 16 = 0\).

Step 4: Interpret the discriminant

The discriminant \(D = 0\) indicates that the quadratic equation has exactly one real solution.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. It is generally expressed in the standard form: \[ ax^2 + bx + c = 0 \] where

• \(a\), \(b\), and \(c\) are coefficients,
• \(a\) is not equal to zero,
• \(x\) represents the variable or the unknown in the equation.

The presence of the term \(ax^2\) classifies it as a quadratic equation, marking it apart from linear equations which lack the squared term.
Quadratic equations can be solved using various methods, including factoring, using the quadratic formula, completing the square, and graphing. However, before finding the solutions, it's useful to determine the nature and number of the solutions using the discriminant.

Real Solutions

In the context of quadratic equations, solutions refer to the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). When we talk about real solutions, we specifically mean solutions that are real numbers as opposed to complex numbers.
The discriminant, represented by \(D\), provides key insights into the nature of these solutions:

• If \(D > 0\), the quadratic equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, also known as a repeated or double root.
• If \(D < 0\), there are no real solutions; however, there are two complex solutions.

Thus, the discriminant serves as a powerful tool to quickly ascertain the number and type of solutions, without needing to solve the equation fully.
In our task with the discriminant of 0, it indicated that there is one real solution for the given quadratic equation.

Coefficients

Coefficients are numerical or constant factors associated with the terms in an equation. In a quadratic equation \( ax^2 + bx + c = 0 \), the coefficients \(a\), \(b\), and \(c\) play specific roles:

• \(a\) is called the leading coefficient, as it is multiplied to the highest power of \(x\), which is \(x^2\).
• \(b\) is the coefficient of the linear term, \(x\).
• \(c\) is the constant term, which stands alone without \(x\).

Identifying these coefficients accurately is crucial, especially when calculating the discriminant. For example, in the problem, the coefficients were identified as \(a = 9\), \(b = -4\), and \(c = \frac{4}{9}\).
These values are then used in the discriminant formula \(D = b^2 - 4ac\) to evaluate the nature of the solutions. Proper manipulation of these values determines whether we have real solutions, and if so, how many.