13Quadratics Before we get started let’s review. A quadratic equation is an equation that canbe written in the form ,where a, b and c are real numbers and a cannot equalzero.In this lesson we are going to discuss quadraticinequalities.
14Quadratic Inequalities What do they look like?Here are some examples:
15Quadratic Inequalities When solving inequalities we are trying tofind all possible values of the variablewhich will make the inequality true.Consider the inequalityWe are trying to find all the values of x for which thequadratic is greater than zero or positive.
16Solving a quadratic inequality algebraically We can find the values where the quadratic equals zeroby solving the equation,
17Solving 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.-23
18Solving a quadratic inequality graphically If the inequality is:- TRUE, shade where “0” is.- FALSE, shade where “0” is not.Solution: The intervals
19Example 2:SolveFirst find the zeros by solving the equation,
20Example 2: Solution: The interval makes up the solution set for the inequality
22Solving 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
23Solving 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.
24Quadratic Inequalities Critical NumberCritical NumberMost 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.
25Quadratic Inequalities x2 + 12x + 32 < 0-x2 – 12x – 32 < 0-8 < x < -4x < -8 or x > -4(-8, -4)(- , -8) U (-4, )
26Quadratic Inequalities x2 + 12x + 32 > 0-x2 – 12x – 32 > 0x < -8 or x > -4-8 < x < -4(- , -8] U [-4, )[-8, -4]
27Quadratic Inequalities -(x + 7)2 – 6 < 0(x + 8)2 + 6 < 0everywhereNo where(- , )These parabolas are all or nothing.
28Special Cases: *Double roots *When “a” is negative…
29When “a” is negative… you must divide by a negative to make “a” positive Ex.-x2 – 12x – 32 < 0
31Quadratic Inequalities: Double Roots Critical Number(x – 2)2 < 0(x – 5)2 > 0Only at one placeEverywhere except 5x = 2x = 5(-, 5) U (5, )These parabolas could be all or nothing.
32Example 3:Solve the inequalityFirst find the zeros.
33Example 3: But these zeros , are complex numbers. What does this mean? Let’s look at the graph of the quadratic,
34Example 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.
35Summary In general, when solving quadratic inequalities Find the zeros by solving the equation you get when you replace the inequality symbol with an equals.Find the intervals around the zeros by graphing it in your calculatorThe solution is the interval or intervals which make the inequality true.
36Tic-Tac-Know: Complete 3 in a row, your choice! Name: ________________