ࡱ>  FG` \+bjbjss 7"O  44485<T7 l48@4:@:@:@JJJAlClClClClClCl$nh2qgl JJ"JJJgl:@:@ |l T T TJj:@ :@Al TJAl T TRUg i:@(8 :|:54[KIhEjl0lghqNq<iq iJJ TJJJJJglglS^JJJlJJJJ )|3d |3   Section V: Quadratic Equations and Functions  Module 1: Solving Quadratic Equations Using Factoring, Square Roots, Graphs, and Completing-the-Square DEFINITION: A quadratic equation is an equation of the form  EMBED Equation.3  where a, b, and c are real numbers and  EMBED Equation.3 .  In your introductory algebra course, you should have solved quadratic equations using factoring, graphs, the quadratic formula, and the square root method. In this module, we will review solving quadratic equations using factoring, using graphs, and using the square root method, as well as introduce solving quadratic equations by completing the square. In the next module we will study solving quadratic equations using the quadratic formula.  SOLVING QUADRATIC EQUATIONS BY FACTORING  example: Solve the following quadratic equations by factoring. a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  SOLUTIONS:  EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4   EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4   EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4   SOLVING QUADRATIC EQUATIONS USING THE SQUARE ROOT METHOD  example: Solve the following quadratic equations using the square root method. a.  EMBED Equation.3  c.  EMBED Equation.3  b.  EMBED Equation.3  d.  EMBED Equation.3  SOLUTIONS:  EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4 .  EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4 .  EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4 .  EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4 .   example: Let  EMBED Equation.3  and  EMBED Equation.3 . Find all x for which  EMBED Equation.3 . SOLUTION:  EMBED Equation.DSMT4  This isnt a quadratic equation. In fact, it is a rational equation. But in order to solve such an equation, we need to clear the fractions and, once we clear the fractions, we have a quadratic equation:  EMBED Equation.DSMT4  Thus, the solution set is  EMBED Equation.DSMT4 .  SOLVING QUADRATIC EQUATIONS USING GRAPHS  example: Use the graph below to solve the quadratic equation  EMBED Equation.DSMT4 .  The graph of  EMBED Equation.DSMT4 . SOLUTION: Since we are given the graph of  EMBED Equation.DSMT4  and we are asked to solve  EMBED Equation.DSMT4 , we need to look for any places on the graph where  EMBED Equation.3 , i.e., we need to find the x-intercepts of the graph. Since the x-intercepts are  EMBED Equation.3  and  EMBED Equation.3 , we can conclude that the solutions are  EMBED Equation.3  or  EMBED Equation.DSMT4 , so the solution set is  EMBED Equation.DSMT4 .  example: Use the graphs below to solve the given quadratic equations. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4    The graph of  EMBED Equation.DSMT4 . The graph of  EMBED Equation.DSMT4 . c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4    The graph of  EMBED Equation.DSMT4 . The graph of  EMBED Equation.DSMT4 . SOLUTIONS: a. Since we are given the graph of  EMBED Equation.DSMT4  and we are asked to solve  EMBED Equation.DSMT4 , we need to look for any places where the graph of  EMBED Equation.DSMT4  intersects the line  EMBED Equation.3 , i.e., we need to find the x-intercepts of the graph. Since the x-intercepts are  EMBED Equation.3  and  EMBED Equation.3 , we can conclude that the solutions are  EMBED Equation.3  or  EMBED Equation.DSMT4 , so the solution set is  EMBED Equation.DSMT4 . b. Since the graph of  EMBED Equation.DSMT4  intersects the line  EMBED Equation.3  only when  EMBED Equation.3 , we can conclude that the solution set is  EMBED Equation.DSMT4 . c. Since the graph of  EMBED Equation.DSMT4  never intersects the line  EMBED Equation.3  (i.e., the x-axis), we see that " EMBED Equation.DSMT4 " never equals 0, so we can conclude that the equation  EMBED Equation.DSMT4  does not have any real solutions. (Note that we can use the quadratic formula or completing-the-square to find the complex numbers that solve the equation.) d. Here, we are given the graph of  EMBED Equation.DSMT4  so we need to first obtain the expression " EMBED Equation.DSMT4 " in the given equation  EMBED Equation.DSMT4  which we can do by solving this equation for 0:  EMBED Equation.DSMT4  We can solve this equation by looking for the x-values where the graph of  EMBED Equation.DSMT4  intersects the line  EMBED Equation.3  (i.e., the x-axis). Thus, the solution set is  EMBED Equation.DSMT4   SOLVING QUADRATIC EQUATIONS BY COMPLETING-THE-SQUARE Using the square root method to solve a quadratic equation only works if we can write the quadratic equation so that one side is the square of a binomial. To write the square of a binomial, we start with a perfect square trinomial, i.e., a trinomial that can be factored into two identical binomial factors). Perfect Square Trinomials(Factored Form EMBED Equation.3 ( EMBED Equation.3  EMBED Equation.3 ( EMBED Equation.3  Notice that for each perfect square trinomial, the constant term of the trinomial is the square of half the linear term (in this case, the coefficient of x). For example,  EMBED Equation.3   EMBED Equation.3    EMBED Equation.3  and  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  The process of writing a quadratic equation so that one side is a perfect square trinomial is called completing the square.  example: Solve  EMBED Equation.3  for x by completing the square. SOLUTION: First, we need to find the square of half the coefficient of the linear term (in this case, the coefficient of x):  EMBED Equation.3  and  EMBED Equation.3  Now we add and subtract 1 on the left-hand side of the equation. Notice that we are just adding zero in a disguised form, so we wont change the solution to the quadratic equation.  EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4 .   example: Solve  EMBED Equation.3  for p by completing-the-square. SOLUTION: First, we need to find the square of half the coefficient of the linear term (in this case, the coefficient of the term  EMBED Equation.DSMT4 ).  EMBED Equation.3  Now we add and subtract 9 on the left-hand side of the equation.  EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4 . To Solve a Quadratic Equation by Completing-the-Square: 1. Write the equation in standard form ( EMBED Equation.3 ). 2. Divide both sides of the equation by the coefficient of the quadratic term, a, in order to make that coefficient 1. 3. Complete the square: a. Multiply the coefficient of the linear term by one-half, then square the result. b. Add and subtract the value obtained in step a. 4. Write the square of a binomial plus a constant, then move the constant to the other side of the equation. 5. Use the square root method to find the solution.  example: Solve  EMBED Equation.3  by completing-the-square. SOLUTION: First, we must divide both sides of the equation by 2 (the coefficient of  EMBED Equation.DSMT4 ) in order to make 1 the coefficient of  EMBED Equation.DSMT4 .  EMBED Equation.3  Next, we find the square of half the coefficient of the linear term (in this case, the coefficient of x):  EMBED Equation.3  Now add and subtract  EMBED Equation.DSMT4  from the left-hand side and solve for x:  EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4 .   example: Solve  EMBED Equation.3  by completing the square. [Recall that we tried to solve this equation graphically and concluded that there are no real solutions. Thus, we should expect that our solutions will be complex numbers.] SOLUTION: First, we need to find the square of half the coefficient of the linear term (in this case, the coefficient of x):  EMBED Equation.3  Now add and subtract  EMBED Equation.DSMT4  on the left-hand side of the equation, and solve for x.  EMBED Equation.3  Thus, the solution set is  EMBED Equation.DSMT4 .   Try these yourself and check your answers. Solve by completing the square. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  SOLUTIONS:  EMBED Equation.DSMT4  Thus, the solution set is  EMBED Equation.DSMT4 .  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