Lesson HOW TO - solve quadratic inequalities

Source code of 'HOW TO - solve quadratic inequalities'

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INTRODUCTION

The secret to being able to do quadratic inequalities is in visualising the graph of the quadratic. Once we have that in our minds (or on the page --> NOT on a calcualtor!), then we can instantly see the answer.

The crux of sketching any graph is in knowing the roots... those points where the curve crosses the x-axis. This is the standard problem of solving any quadratic.

Inequalities merely ask you to quote the values of x where the graph is either:

above the x-axis (where y>0), or
below the x-axis (where y<0)

EXAMPLES

Q Solve {{{x^2 + 3x + 2 < 0}}}
A First, solve {{{x^2 + 3x + 2 = 0}}}, to find the roots, i.e. where the curve crosses the x-axis.

answer for this is at x=-1 and x=-2, so you should then be imagining the graph as {{{graph(200,200, -3,0,-1,4, x^2 + 3x + 2)}}}.

Everything so far is "normal" quadratic equations. Now we do the inequality part, once we have the sketch of the curve in our heads.

The question wants to know "which x-values do we need that will give a quadratic function value less than zero".
--> This is all x-values between x=-2 and x=-1 --> we write this as -2 < x < -1.

Q Solve {{{x^2 + 3x + 2 > 0}}}
A Again, solve the "equal" version first, to find the roots...those points where the curve crosses the x-axis. Sketch it and then do the "inequality" part: The answer would be the 2 outer regions.

--> hence x< -2 and x>-1