Algebraic Concepts of Quadratics
Unit Objectives
Learners will work with various equations finding zeros. By completing the square and using the quadratic formula. Learners will also use documents in the TI-Nspire to better understand both methods.
Accessing Prior Knowledge
In order to solve quadratic equations you must remember what "perfect squares" are. A perfect square is a trinomial that can be factored into two binomials that are identical. These are called perfect square because if we think of multiplication as an area model the two binomials would have the same length. Hence the figure would be a square. Some examples of perfect squares with their factored forms are as follows:
- x2 + 4x + 4 = (x + 2)(x + 2)
- x2 + 8x + 16 = (x + 4)(x + 4)
- x2 + 20x + 100 = (x + 10)(x + 10)
- x2 - 18x + 81 = (x - 9)(x - 9)
- x2 + 10x +25 = (x + 5)(x + 5)
- x2 - 22x + 121 = (x - 11)(x - 11)
Solving Quadratics by Completing the Squares
Finding What is Needed for a Perfect Square
For the following problems find what value of n is needed to make the following a perfect square.
- x2 - 12x + n
- x2 + 18x + n
- x2 - 4x + n
- x2 - 9x + n
To check your answers click here
Completing the Square
When completing the square you must first make one side of the equation a perfect square. When doing this you must remember the Golden Rule: "Do unto one side of an equation as you would do unto the other."
- x2 + 8x = 20
- x2 + 8x + 42 = 20 + 42
- x2 + 8x + 16 = 36
When one side of an equations is in the form of a perfect square you can factor the trinomial into one binomial squared.
- (x + 4)2 = 36
Taking the square root of both sides of the equation and solving for x you will obtain the roots of the equation.
- x + 4 = +-6
- x = -4 +-6
- x = 2 and -10
Extra Practice
Complete these following examples of increasing difficulty. As you are completing these examples think of the following ideas:
- Creating perfect squares.
- Do all answers have to be rational?
- What happens when the x2 has a coefficient other than 1?
- x2 - 12x = -9
- x2 + 9x = -7
- 2x2 + 8x = 20
Answers to the above problems can be found by clicking this link Solution to Completing the Square.
Solving Quadratics using the Quadratic Formula
The Quadratic Formula
Completing the square is one way of solving quadratic equations. There is also another; the quadratic formula. The quadratic formula can be used to solve quadratic equations but it first must be memorized. To aid in the memorization of the quadratic formula it can be helpful to put it to a familiar tune.
- For an example click the following link:The Quadratic Formula Song
How the Quadratic Formula Works
The quadratic formula is a tool that allows you to find the roots of a quadratic equation much like the method of completing the square. In this method you must first arrange the equation into standard form: ax2 + bx + c = 0. When arranging a quadratic equation into standard form it becomes very easy to identify the a, b, and c. Let us now find the roots of the previous example see how these two methods compare.
- Note due to increase of mathematical notation please click on the link below to continue with the quadratic equation examples.
Modeling with the TI-Nspire
After solving the previous 4 examples. You should realize that both methods are perfectly acceptable for finding roots. Graphically speaking what did we just do? By looking at the equations following equations we can better understand the applications of the quadratic formula visually.
- 0 = x2 + 8x - 20
- y = x2 + 8x - 20
The following TI-Nspire document you can examine the similarities between finding the roots of an equation the zeros of the corresponding graph.
Self-Reflection
Now that you have solved quadratic equations using the quadratic formula and by completing the square, take a few moments and reflect on the following questions.
- Are there advantages or disadvantages to the two methods discussed.
- Keeping the above question in mind, is there a time when one method is more practical than the other?
- What are the limitations of the TI-Nspire graphing capabilities?
- Does the TI-Nspire support the two algebraic means discussed in this lesson? How?
Back to home page Quadratic Modeling with the TI-Nspire
Advance to the next lesson: Two Different Forms of Quadratic Equations
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