Lesson summary for Quadratic Inequalities
- Quadratic inequalities involve expressions like $ax^2 + bx + c < 0$, $ax^2 + bx + c > 0$, $ax^2 + bx + c \leq 0$, and $ax^2 + bx + c \geq 0$.
- They describe the values of ( x ) that satisfy these conditions and can be visualized on a number line or coordinate plane.
- Solving Using Product Properties:
- For $ax^2 + bx + c > 0$:
- Rewrite the inequality in the form $mn > 0$, where ( m ) and ( n ) are factors.
- $mn > 0$ if:
- $m > 0$ and $n > 0$, or
- $m < 0$ and $n < 0$.
- $mn > 0$ if:
- For $ax^2 + bx + c < 0$:
- Rewrite in the form $mn < 0$.
- $mn < 0$ if:
- $m > 0$ and $n < 0$, or
- $m < 0$ and $n > 0$.
- Solving Using Sign Charts:
- Steps:
- Rewrite the inequality with ( 0 ) on the right-hand side.
- Factor the left-hand side if possible.
- Identify roots, which divide the number line into intervals.
- Test each interval and create a sign chart.
- Determine where the product is positive or negative.
- Steps:
- In summary, solving quadratic inequalities involves rewriting them in factored form, applying product properties, or using sign charts to determine the intervals where the inequality holds.