The discriminant of a quadratic equation, ax2 + bx + c = 0 is D = b2 – 4ac. If D>0, the quadratic equation has two distinct roots; if D<0, then the equation has no real roots; and, if D=0, the we have two equal roots. Let’s apply it in the following problem. Continue reading “Application of the Discriminant”
Tag: quadratic equations
Solving quadratic equations by completing the square
I’m not a fan of teaching the quadratic formula for solving the roots of quadratic equations because the sight of the outrageous formula itself is enough to make students wish they are invisible in their algebra class. Indeed who wants to have to do with
Of course not all quadratic equations can be solved by factoring. Here’s how I try to resolve the situation. Before quadratics, students have been solving linear equations. So if you ask them to solve 
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Students will try to guess and check until they find the values of x that will make the equation true. They will continue to use this technique until you give them something like 
Clearly the left hand side is not a square. The way to make one is to cut-off half of the 4x area. But it makes an incomplete square!
Let’s complete it by adding a 2 by 2 square. To keep the balance we add the same amount on the right hand side.
It should be now easy solving for x by extracting the root and using the properties of equality.
I believe that this process will make sense more than using the quadratic formula. Students just memorize the formula without understanding. They also will not remember a piece of it the next day anyway. I’m not saying the quadratic fomula is not completely useful. One application of it is on using the Cosine Rule for ambiguous case.
Should the method of factoring be taught first? I believe it’s best to introduce the students to the method of completing the square first (with the visuals, of course). Once the students get the hang of this procedure, the first thing that they will drop is drawing the rectangle and square and just do it mentally.You can later ask them to investigate the structure of quadratic equations where it is no longer necessary to transfer the constant on the other side. Solving quadratic equation by factoring therefore is a shortcut students should deduce from the procedure of completing the square.
Any new procedure should be linked to previously learned procedure or it should be an improvement of the first. This is my reason why I think the process I described above is a natural sequence to the process of solving linear equation that students already learned. Another reason is that most of the problems students encounter involving quadratic equation is of the form 
I also developed a geogebra applets Completing the Square Solver and Quadratic Equation Solver that I posted in AgIMat. You can use them to solve quadratic equations and to investigate their roots.
What exactly is the square root of 36?
Mathematics is already a difficult subject so let us not make it more difficult to students to make sense of with the confusing rules we tell them. For example, we insist that the square root of 36 is 6 and -6 yet we also insist that ?36 is equal to 6 and not -6. A colleague said that by convention, when you ask ‘what is the square root of 36?’ the answer is 6 and -6 but when you just write the symbol ‘?36 =?’ the answer must be 6. So, I said, do you mean that if you just want the positive root, you do not read the symbol ?36 otherwise it will have 2 values? Here’s an excerpt from an article I read recently about the matter. It’s title is “What root do you want to take?” by Derek Ball.
Teachers and books make the most extraordinary statements about quadratic equations. All quadratic equations have two solutions, they say. How about x2 – 8x + 16 = 0? Does that have two solutions? “Yes”, they say, “it has the repeated solution 4. If you put x = 4 the equation is true.” Fine. “And if you put x=4 …” Yes, I heard you the first time.
Of course, I can mock, but it is quite useful for some purposes to think of this equation having a ‘repeated solution’, just as it is quite useful for other purposes to think of it as having just one solution. Confused? So you should be. To add to the confusion I shall ask you this question: ‘What is the square root of 9?’ Depending on context you might answer ‘3 or -3’. If I ask you: ‘What is the square root of zero’, what are you to answer: ‘Zero’ or ‘Zero and zero’ or ‘Zero and minus zero’?
Anyway, solving quadratics is where things like ?36 really come into their own. If you want to solve x2 – x – 1 = 0, you can use ‘the formula’ and obtain the solution x = (1 + ?5)/2. What I need to remember when interpreting this solution is that ?5 is positive. Or do I? Why do I? And if every collection of symbols is supposed to represent (at most) one number, what about‘+?5’?
And sooner or later you may want to solve equations like z2 – 4z + 5 = 0 and you may perhaps use the quadratic formula and obtain z = (4 +(?4))/2. Now new questions arise. Am I allowed to write ?(-4) and if I am what is its value? Is it 2i or -2i and why? Perhaps you want to say the answer is obviously 2i. In that case how about a quadratic equation whose solution involves ?(3 – 4i). Is this – 2 + i or is it 2 – i? What I am saying is that we use symbols to help us solve problems. If we use + in front of a square root sign this reminds us that in order to solve the quadratic equation completely we need to remember to take two different values for the square root. Knowing that ?36 means 6 and not -6 does not matter at all, unless we are asked silly questions in pub quizzes or GCSE exams like ‘What is ?36?’ and we are supposed (for some unknown reason) to know that we have to answer 6 and not -6.
Still not convinced? Well, am I allowed to write 3?-27 and is its value allowed to be -3? Its value could be other things too, of course. Does that complicate things? Does that make us think that we can only judge what a symbol means from the context? So, far be it from me to defend textbooks, but perhaps they have some justification for using the square root sign inconsistently. As for fractional powers, they seem to raise exactly the same issues as root signs.The moral of this tale is surely that we move away from asking questions towhich we want a correct answer, so thatwe can say ‘Right’ or ‘Wrong’, and instead solve problems that interest us, talk about mathematics and connect ideas together. And I hope we sometimes get confused, because confusion is often a spur to sorting our ideas out.
Amen to that. For more confusing rules that we give to our students read my post on Mistakes vs. Misconceptions. You may also want to know the more about Algebra Errors.