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**SLIDE SHOW INSTRUCTIONS **

This presentation is completely under your control. This lesson will show only one step at a time, to see the next step you must press a key. (Actual names written on a key are in green) TO STOP THE SLIDE SHOW: press ‘escape’ (Esc, top left of keyboard) TO MOVE FORWARD: press the “spacebar” or Enter (PageDn, , , also work) TO MOVE BACKWARD: press the key (PageUp, or also work)

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**This tutorial is in two parts:**

Part One walks you through the procedures for solving a quadratic equation using the Quadratic Formula. Part Two describes the process of predicting the Type of Solution you will get without actually having to use the whole formula. There is a chart at the end of the lesson which is a guide for Predicting Types of Solutions.

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The Quadratic Formula

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**where a, b and c are numbers II. How to use the Quadratic Formula:**

I. When to use the Quadratic Formula: When factoring any equation that is in Quadratic Form Quadratic Form: ax2 + bx + c = 0 where a, b and c are numbers OVERVIEW II. How to use the Quadratic Formula: Replace each a, b and c in the formula below with it’s value, then simplify it as far as possible. x =

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**2x2 - 5x + 7 = 0 Here’s a tip to prevent sign errors: Example 1:**

The coefficient of the x2 term is the “a” The coefficient of the x term is the “b” The constant is the “c” Here’s a tip to prevent sign errors: First write the Quadratic Formula with empty parentheses in place of each a, b and c. x = -( ) - 4( )( ) 2( ) b a c Notice: The “x”, the “4” and the “2” never change. Also, notice where the two negatives and the “plus/minus” sign are located. Lastly, notice that the first term inside the square root sign is squared (the 2 is outside the parentheses).

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**The first term becomes a +5 and the bottom is a 4**

2x2 - 5x + 7 = 0 Example 1: Fill in the empty parentheses with the correct numbers (including the sign) and simplify the equation x = -(-5) 2(2) b a c The first term becomes a +5 and the bottom is a 4 x = 5 4

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**5 x = 4 We can simplify the square root piece on the side,**

this will save us a lot of writing. x = 5 4 This section of the formula (the square root piece) is called the discriminant. Now we can put this back into the fraction

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**5 x = 4 Look at the numerator (the top of the fraction),**

The 5 is a real number and the discriminant is imaginary. This type of number is called a Complex Number. This means the numerator can’t be simplified any further. x = 5 4 The equation is now in simplified form. (i.e. this is the answer) But, finding the value of x is not the only question you might be asked to answer. You may be asked to describe the type of solution you will get for x.

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**x = 5 x = 5 + x = 5 - and The solution to this equation is: 4 4 4**

Since the fraction has a “” symbol in it, it is really two numbers. x = 5 + 4 x = 5 - 4 and Type of Solution: Two Complex Solutions were found for the quadratic equation 2x2 - 5x + 7 = 0

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**Write the equation with empty parentheses for a, b and c**

4x2 + 7x - 2 = 0 Example 2: Write the equation with empty parentheses for a, b and c x = -( ) - 4( )( ) 2( ) b a c Fill in the empty parentheses with the correct numbers (including the signs) and simplify the equation x = -(7) 2(4) b a c

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**The first term becomes a -7 and the bottom is an 8**

x = -7 8 Now simplify the discriminant Now we can put this back into the fraction

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**The solution to this equation is:**

x = -7 9 8 Since the fraction has a “” symbol in it, this is 2 numbers. x = -7 + 9 8 x = -7 - 9 8 and The numerator has two real numbers which can be combined. and Type of Solution: Two Real Number Solutions were found for the quadratic equation 4x2 + 7x - 2 = 0

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**Write the equation with empty parentheses for a, b and c**

3x2 - x - 6 = 0 Example 3: Notice the “b” is a -1 Write the equation with empty parentheses for a, b and c x = -( ) - 4( )( ) 2( ) b a c Fill in the empty parentheses with the correct numbers (including the signs) and simplify the equation x = -(-1) 2(3) b a c

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**The first term becomes a 1 and the bottom is a 6**

x = 1 6 Now simplify the discriminant Now we can put this back into the fraction

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**x = 1 x = 1 + x = 1 - and The solution to this equation is: 6 6 6**

Since the fraction has a “” symbol in it, this is 2 numbers. x = 1 + 6 x = 1 - 6 and The numerator has two real numbers which cannot be combined. Type of Solution: Two Real Number Solutions were found for the quadratic equation 3x2 - x - 6 = 0

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**Write the equation with empty parentheses for a, b and c**

4x2 + 6x + 1 = 0 Example 4: Write the equation with empty parentheses for a, b and c x = -( ) - 4( )( ) 2( ) b a c Fill in the empty parentheses with the correct numbers (including the signs) and simplify the equation x = -(6) 2(4) b a c

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**-6 x = 8 The first term becomes a 1 and the bottom is a 6**

Now simplify the discriminant Now we can put this back into the fraction

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**x = -6 x = -6 + x = -6 - and x = -3 + x = -3 - and**

The solution to this equation is: x = -6 8 Since the fraction has a “” symbol in it, this is 2 numbers. x = -6 + 8 x = -6 - 8 and The numerator has two real numbers which cannot be combined. But, they can be REDUCED, each of these terms can be divided by two. x = -3 + 4 x = -3 - 4 and Type of Solution: Two Real Number Solutions were found for the quadratic equation 4x2 + 6x + 1 = 0

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**Write the equation with empty parentheses for a, b and c**

4x2 + 4x + 1 = 0 Example 5: Write the equation with empty parentheses for a, b and c x = -( ) - 4( )( ) 2( ) b a c Fill in the empty parentheses with the correct numbers (including the signs) and simplify the equation x = -(4) 2(4) b a c

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**-4 x = 8 The first term becomes a 1 and the bottom is an 8**

Now simplify the discriminant Now we can put this back into the fraction

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**-4 0 x = and The solution to this equation is: 8 Type of Solution:**

Since the fraction has a “” symbol in it, this is 2 numbers. and This is one real number repeated twice. So, once we reduce it, the solution is: Type of Solution: One Real Number Solution was found for the quadratic equation 4x2 + 4x + 1 = 0

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Let’s do a recap: We have seen many examples of how to use the quadratic formula. The procedure was the same in each case: 1. We wrote the quadratic formula using empty parentheses in place of the a, b and c. 2. We filled in the values of a, b and c, including the signs. 3. We simplified the equation. 4. We split the simplified equation into two pieces to determine if it could be simplified further. Next, we described the type of solution(s) we obtained. We got one of the following three types of solutions: 1. Two real number solutions. 2. Two complex number solutions. 3. One real number solution.

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**Predicting the Type of Solution**

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**Example 1 had the solution:**

The type of solution is depends on the value of the discriminant. We can predict the type of solution by looking at the discriminant before or after we simplify or reduce it. Go back and look at all the examples we worked. Example 1 had the solution: Discriminant is negative (i.e. imaginary). Predict the Type of Solution: Two Complex Solutions Actual Solutions:

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**Example 2 had the solution:**

The numerator can be simplified further Discriminant is positive (i.e. Real). Predict the Type of Solution: Two Real Number Solutions Actual Solutions:

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**Example 3 had the solution:**

It won’t reduce Discriminant is positive (i.e. Real). Predict the Type of Solution: Two Real Number Solutions Actual Solutions:

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**Example 4 had the solution:**

The fraction can be simplified further Discriminant is positive (i.e. Real). Predict the Type of Solution: Two Real Number Solutions Actual Solutions:

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**Example 5 had the solution:**

This becomes a single fraction Discriminant is zero (i.e. Real). Predict the Type of Solution: One Real Number Solution Actual Solution:

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**Two Real Number Solutions**

The chart below outlines how to predict the type of solution(s) by looking at the discriminant alone. Discriminant Produces Type of Solution positive real numbers Two Real Number Solutions negative imaginary numbers Two Complex Number Solutions zero one real number One Real Number Solution

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**[Two Real, Two Complex or One Real Number Solution(s)]**

Practice Exercises Part I: Predict the types of solutions you will get for the following equations. [Two Real, Two Complex or One Real Number Solution(s)] Equations Predictions 1. 3x2- 2x - 1 2. 4x2+ 2x - 5 3. 2x2+ 3x + 4 4. 9x2 - 6x + 1 5. 3x2 + 5x + 2 6. 5x2- 2x + 1 2 Reals 2 Complex 1 Real 2 Real Part II: Use the Quadratic Formula to find the actual solutions for the six equations above. [See if your predictions were correct] Solutions: , -1/ /3 /3, -1 (completely reduced)

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**Questions? send e-mail to: lgreene1@satx.rr.com**

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