Quadratics Before we get started let’s review. A quadratic equation is an equation that can be written in the form, where a, b and c are real numbers and a cannot equal zero. In this lesson we are going to discuss quadratic inequalities.
Quadratic Inequalities What do they look like? Here are some examples:
Quadratic Inequalities When solving inequalities we are trying to find all possible values of the variable which will make the inequality true. Consider the inequality We are trying to find all the values of x for which the quadratic is greater than zero or positive.
Solving a quadratic inequality algebraically We can find the values where the quadratic equals zero by solving the equation,
Solving a quadratic inequality algebraically For the quadratic inequality, we found zeros 3 and –2 by solving the equation. *To find the solution set you must test a value. ALWAYS USE “0” to determine where to shade. -2 3
Solving a quadratic inequality graphically If the inequality is: - TRUE, shade where “0” is. - FALSE, shade where “0” is not. Solution: The intervals
Example 2: Solve First find the zeros by solving the equation,
Example 2: Solution: The interval makes up the solution set for the inequality.
Solving a quadratic inequality graphically You may recall the graph of a quadratic function is a parabola and the values we just found are the zeros or x-intercepts. The graph of is
Solving a quadratic inequality graphically From the graph we can see that in the intervals around the zeros, the graph is either above the x-axis (positive or > 0) or below the x-axis (negative or <0). So we can see from the graph the interval or intervals where the inequality is positive.
Quadratic Inequalities Most parabolas can be broken up into 3 sections: 2 outer sections and 1 inner section. A solution set for a quadratic inequality will be either the 2 outer sections or the 1 inner section. Critical Number
Quadratic Inequalities x 2 + 12x + 32 < 0-x 2 – 12x – 32 < 0 (- , -8) U (-4, )) (-8, -4) -8 < x < -4 x -4
Quadratic Inequalities (- , -8] U [-4, ) [-8, -4] x 2 + 12x + 32 > 0-x 2 – 12x – 32 > 0 x -4 -8 < x < -4
Quadratic Inequalities -(x + 7) 2 – 6 < 0(x + 8) 2 + 6 < 0 No where (- , ) everywhere 0 These parabolas are all or nothing.
Special Cases: *Double roots *When “a” is negative…
When “a” is negative… you must divide by a negative to make “a” positive Ex. -x 2 – 12x – 32 < 0
Quadratic Inequalities: Double Roots (x – 5) 2 > 0 Everywhere except 5 x = 2 Only at one place x = 5 (x – 2) 2 < 0 (- , 5) U (5, )  These parabolas could be all or nothing. Critical Number
Example 3: Solve the inequality. First find the zeros.
Example 3: But these zeros, are complex numbers. What does this mean? Let’s look at the graph of the quadratic,
Example 3: We can see from the graph of the quadratic that the curve never intersects the x-axis and the parabola is entirely below the x-axis. Thus the inequality is always true.
Summary In general, when solving quadratic inequalities 1. Find the zeros by solving the equation you get when you replace the inequality symbol with an equals. 2. Find the intervals around the zeros by graphing it in your calculator 3. The solution is the interval or intervals which make the inequality true.
Tic-Tac-Know: Complete 3 in a row, your choice! Name: ________________