Problem 35 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ 9 x^{2}-6 x+1 \leq 0 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Quadratic

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

Step 2: Determine the Discriminant

The discriminant \( \Delta \) is given by \( b^2 - 4ac \). For our quadratic, \( \Delta = (-6)^2 - 4 \cdot 9 \cdot 1 = 36 - 36 = 0 \), which implies the quadratic has a repeated real root.

Step 3: Solve for Roots

Since the discriminant is zero, there is one real repeated root. Use the quadratic formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \). This gives \( x = \frac{-(-6) \pm 0}{2 \cdot 9} = \frac{6}{18} = \frac{1}{3} \).

Step 4: Write the Factored Form

Given the root \( x = \frac{1}{3} \) and the fact that it is repeated, the quadratic can be factored as \( (3x - 1)^2 = 0 \). This matches our quadratic equation since expanding gives back the original equation.

Step 5: Analyze the Inequality

Since \( (3x - 1)^2 \geq 0 \) for all real numbers and \( = 0 \) only when \( 3x - 1 = 0 \) or \( x = \frac{1}{3} \), the inequality \( (3x - 1)^2 \leq 0 \) is only satisfied at \( x = \frac{1}{3} \).

Step 6: Conclusion

The solution to the inequality \( 9x^2 - 6x + 1 \leq 0 \) using algebraic methods is \( x = \frac{1}{3} \). This is the only point where the parabola touches the x-axis, making it the solution.

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two. They are of the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a eq 0 \). These functions graph into a shape called a parabola, which can open upwards or downwards.
The position and direction of the parabola are heavily influenced by the values of \( a, b, \) and \( c \):

• If \( a > 0 \), the parabola opens upwards like a U.
• If \( a < 0 \), it opens downwards like an upside-down U.
• The vertex of the parabola, which is its highest or lowest point, can be found using the formula \( x = -\frac{b}{2a} \).

Quadratics also have a specific symmetry around a vertical line called the axis of symmetry. This axis passes through the vertex.

Discriminant

The discriminant of a quadratic equation, given by the formula \( \Delta = b^2 - 4ac \), plays a critical role in determining the nature of the roots of the quadratic function.
Considering the discriminant can tell you:

• When \( \Delta > 0 \), the quadratic has two distinct real roots.
• When \( \Delta = 0 \), there is exactly one real root, known as a repeated or double root.
• When \( \Delta < 0 \), the roots are complex and not real.

In the context of solving quadratic inequalities, knowing the discriminant helps decide how the roots affect the inequality. For example, a significant \( \Delta = 0 \) indicates where the vertex touches the x-axis.

Factored Form

The factored form of a quadratic function expresses it as the product of two linear expressions. For example, a quadratic \( ax^2 + bx + c \) can be written as \( a(x - r_1)(x - r_2) \) where \( r_1 \) and \( r_2 \) are the roots.
Factoring a quadratic can simplify the process of solving equations and inequalities.
For the quadratic \( 9x^2 - 6x + 1 \), we found the repeated root \( x = \frac{1}{3} \) using the discriminant. This means it factors into \((3x - 1)^2\), reflecting the vertex's touchpoint on the graph.
Understanding this form also allows for easier analysis of the quadratic inequality because it highlights the relationship between the quadratic expression and its roots.

Roots of Quadratic

Roots of a quadratic are essentially the x-values where the quadratic function crosses or touches the x-axis. They are found using various methods, one of which is the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
Where \( \Delta = b^2 - 4ac \) is the discriminant of the quadratic.
Another approach is factoring, when possible, to solve for the roots directly. For our quadratic inequality \( 9x^2 - 6x + 1 \leq 0 \), solving \((3x - 1)^2 = 0 \) yields the single root \( x = \frac{1}{3} \).
The roots provide pivotal information in graphing and solving quadratic inequalities, determining where or if the parabola intersects the x-axis. In context, knowing there’s a single repeated root guided the conclusion that \( x = \frac{1}{3} \) satisfies the inequality.