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Magazine Chapter 6: Problem 34
Solve each inequality. $$9 x^{2}-6 x+1 \leq 0$$
Short Answer
Expert verified
The solution is \( x = \frac{1}{3} \).
Step by step solution
01
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 \( c = 1 \).
02
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.
03
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 \( x = \frac{1}{3} \).
04
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 \( x = \frac{1}{3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
- \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
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.
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:
To find this double root, we use the formula:
- \( x = \frac{-b}{2a} \)
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.
