2-3 2 Quadratic Inequalities Explained
Key Concepts of Quadratic Inequalities
Quadratic inequalities are inequalities that involve quadratic expressions, which are polynomials of the second degree. These inequalities can be solved by analyzing the graph of the quadratic function, identifying the roots, and determining the intervals where the inequality holds true.
1. Understanding Quadratic Inequalities
A quadratic inequality is typically written in the form \( ax^2 + bx + c < 0 \) or \( ax^2 + bx + c > 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \). The inequality sign can also be \( \leq \) or \( \geq \).
2. Solving Quadratic Inequalities
To solve a quadratic inequality, follow these steps:
- Find the roots of the quadratic equation \( ax^2 + bx + c = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
- Determine the intervals created by the roots on the number line.
- Test each interval to see where the inequality holds true.
3. Examples and Analogies
Example 1:
Solve the inequality \( x^2 - 4x + 3 > 0 \):
1. Find the roots of \( x^2 - 4x + 3 = 0 \):
\[ x = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2} \]
\[ x = 3 \quad \text{or} \quad x = 1 \]
2. The roots divide the number line into three intervals: \( (-\infty, 1) \), \( (1, 3) \), and \( (3, \infty) \).
3. Test each interval:
For \( x < 1 \): Choose \( x = 0 \), \( 0^2 - 4(0) + 3 = 3 > 0 \).
For \( 1 < x < 3 \): Choose \( x = 2 \), \( 2^2 - 4(2) + 3 = -1 < 0 \).
For \( x > 3 \): Choose \( x = 4 \), \( 4^2 - 4(4) + 3 = 3 > 0 \).
The solution is \( x \in (-\infty, 1) \cup (3, \infty) \).
Example 2:
Solve the inequality \( -x^2 + 5x - 6 \leq 0 \):
1. Find the roots of \( -x^2 + 5x - 6 = 0 \):
\[ x = \frac{-5 \pm \sqrt{25 - 24}}{-2} = \frac{-5 \pm 1}{-2} \]
\[ x = 3 \quad \text{or} \quad x = 2 \]
2. The roots divide the number line into three intervals: \( (-\infty, 2) \), \( (2, 3) \), and \( (3, \infty) \).
3. Test each interval:
For \( x < 2 \): Choose \( x = 1 \), \( -1^2 + 5(1) - 6 = -2 \leq 0 \).
For \( 2 < x < 3 \): Choose \( x = 2.5 \), \( -(2.5)^2 + 5(2.5) - 6 = -0.25 \leq 0 \).
For \( x > 3 \): Choose \( x = 4 \), \( -4^2 + 5(4) - 6 = -2 \leq 0 \).
The solution is \( x \in (-\infty, 2] \cup [3, \infty) \).
4. Practical Applications
Quadratic inequalities are used in various real-world applications, such as optimizing profit functions in business, determining feasible regions in linear programming, and analyzing physical phenomena like projectile motion.