Problem 34 - Algebra for College Students

Question

Solve each inequality. $$9 x^{2}-6 x+1 \leq 0$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{3} \).

Step 1: Recognize the Quadratic Formula

The inequality given is a quadratic inequality: \( 9x^2 - 6x + 1 \leq 0 \). It is in the standard form \( ax^2 + bx + c \leq 0 \). Here, \( a = 9 \), \( b = -6 \), and \( c = 1 \).

Step 2: Solve the Quadratic Equation

To solve the inequality, first solve the corresponding equation \( 9x^2 - 6x + 1 = 0 \) using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 9 \), \( b = -6 \), \( c = 1 \).Calculate the discriminant: \( b^2 - 4ac = (-6)^2 - 4 \times 9 \times 1 = 36 - 36 = 0 \). Since the discriminant is zero, there is exactly one solution to the equation.

Step 3: Find the Double Root

Since the discriminant is zero, there is a double root given by \( x = \frac{-b}{2a} \). Calculating it, we get \( x = \frac{-(-6)}{2 \times 9} = \frac{6}{18} = \frac{1}{3} \). Thus, the double root of the equation is \( x = \frac{1}{3} \).

Step 4: Determine the Solution to the Inequality

For a quadratic inequality \( ax^2 + bx + c \leq 0 \) with a single root \( r \), the inequality holds for the closed interval \([r, r]\) which is simply \( \{r\} \). This means \( x \leq 0 \) only at this particular point. Therefore, the solution is \( x = \frac{1}{3} \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The formula is expressed as:

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

Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation with \( a eq 0 \). This formula allows us to find the roots or solutions of a quadratic equation. The "+" and "-" in the formula give us two possible solutions, illustrating how quadratic equations typically have two roots. However, the nature of these roots depends significantly on the discriminant, a key part of this formula.

Discriminant

The discriminant is an important component of the quadratic formula, defined as \( b^2 - 4ac \). It determines the nature of the roots of the quadratic equation:

• If the discriminant is positive, \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If the discriminant is zero, \( b^2 - 4ac = 0 \), there is exactly one real root, often referred to as a double root.
• If the discriminant is negative, \( b^2 - 4ac < 0 \), the equation has no real roots and instead has two complex roots.

This is why calculating the discriminant is a crucial step when solving quadratic equations or inequalities. It tells us how many and what type of solutions we can expect, guiding our approach to finding them.

Double Root

A double root occurs in a quadratic equation when the discriminant is zero, \( b^2 - 4ac = 0 \). In this case, there is exactly one unique solution, often called a repeated or double root.
To find this double root, we use the formula:

• \( x = \frac{-b}{2a} \)

This simplifies the quadratic equation to a single root because both potential solutions in the quadratic formula converge to the same value. In the context of our exercise, the equation \( 9x^2 - 6x + 1 = 0 \) has a discriminant of zero. Thus, the double root is \( x = \frac{1}{3} \), meaning this identical solution satisfies the equation.

Solving Inequalities

Solving a quadratic inequality, such as \( 9x^2 - 6x + 1 \leq 0 \), involves determining the set of values for \( x \) that satisfy the inequality. With a double root, the inequality solution depends on the nature of the quadratic curve:

• If the inequality is \( \leq \) and there is a double root, \( x = r \), then the inequality holds at this point.
• The solution set is a single value at \( x = r \), resulting in a very limited solution set, such as \( x = \frac{1}{3} \) for our exercise.

The shape of the parabola, especially when \( r \) is a minimum point, confirms that only when \( x \) is exactly at the double root, the inequality holds true. Thus, for this exercise, the solution to \( 9x^2 - 6x + 1 \leq 0 \) is \( x = \frac{1}{3} \).