Showing posts with label quadratic. Show all posts
Showing posts with label quadratic. Show all posts

Quadratic Inequalities

Quadratic inequalities can be solved in a number of ways. We will focus on solving them by graphing first and then by using sign charts.  It is important to understand what solutions to these inequalities look like before learning the quick and easy method for solving them.

Method 1: Solve by Graphing

   Step 1: Place the inequality in standard form with zero on one side.
   Step 2: Graph the quadratic equation.
   Step 3: Shade the x-values that produce the desired results.
   Step 4: Convert the shading to interval notation.

Solve and express the solution set in interval notation.
In the above example we shaded the x-values for which the graph was above the x-axis. If the problem asked us to solve
or for what x-values is the quadratic less than or equal to 0, then the solution would have been all x-values in the interval [1, 3] - the x-values where the graph is below the x-axis. Notice that the numbers on the x-axis x = 1 and x = 3 separate the positive and negative y-values on the graph, these happen to be the x-intercepts.  The x-intercepts or the zeros of a polynomial are called critical numbers.

Solve by graphing.
Remember to use open dots for strict inequalities < or > and closed dots for inclusive inequalities.

Often it is quite tedious and difficult to graph each inequality when trying to solve them.  There is a shorter method and it involves sign charts.  The idea is to find the critical numbers, the x-values where the y-values could change from positive to negative and create a sign chart to determine which intervals to shade on the x-axis.

Method 2: Solve using sign charts.

   Step 1: Place the inequality in standard form with zero on one side.
   Step 2: Find the critical numbers (for quadratics - the x-intercepts.)
   Step 3: Create a sign chart by determining the sign in each interval bounded by the critical numbers.
   Step 4: Use the sign chart to answer the question.

Solve and graph the solution set.
Find the critical numbers by setting the quadratic expression equal to zero and solve.
 Determine the results + or - in each interval bounded by the critical numbers by testing values in each interval.
Use the sign chart to answer the question. In this case we are looking for the x-values that produce negative results as indicated by the inequality < 0 "less than zero" in the original question.

When testing values in the intervals created by the critical numbers the actual value is not necessary, we are only concerned with its sign.  The sign + or - will be the same for any value in the interval so you may choose any number within the interval when testing.
Solve and graph the solution set.
Tip: Save some time and just determine if the corresponding y-value is positive or negative. If the polynomial factors then use the factors to determine if the interval will produce positive or negative y-values. There is no need to find the actual values.
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Solving Equations Quadratic in Form

In this section, we make use of all the techniques that we have learned so far for solving quadratic equations. In fact, the equations found here are reducible to quadratic form.
Here are most of the reducible equations that we are likely to encounter. Begin by trying to identify what can be squared to obtain the leading variable term.
Tip: Look at the middle term for a hint as to what u should be.
In the previous solved problem, we certainly could have distributed the expression on the left side, put the equation in standard form then re-factored it. Instead, here we are illustrating a technique that will be used to easily solve many other equations that are quadratic in form.

Solve by making a u-substitution.

Solve: x^6 + 26x^3 -27
Six Answers: { -3, 1, (3±3iSqrt(3))/2, (-1±iSqrt(3))/2 }

So far we have been able to factor after we make the u-substitution.  If the resulting quadratic equation does not factor, then use the quadratic formula.
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Intermediate Algebra Exam #3

Click on the 10 question exam covering topics in chapter 6. Give yourself one hour to try all of the problems and then come back and check your answers.


9. The height of a baseball in feet tossed upward is given by the function
where t represents time in seconds.  What is the maximum height of the baseball? 
10. Find a quadratic equation with solutions: {±5i}