1. The MEnTe Program Math Enrichment through Technology
Completing the Square
The MEnTe Program Math Enrichment through Technology
Title V East Los Angeles College
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2. The easiest quadratic equations to solve are of the type
where r is any constant.
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3. The solution to is
The answer comes from the fact that we solve the equation by taking the square root of both sides
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4. Since is a constant, we get
where both sides are positive, but since is a variable it could be negative, yet is positive and is also positive.
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5. To get around this contradiction we need to insist that
If we write and is negative, we are saying that a positive number is negative, which it cannot be.
To get around this contradiction we need to insist that
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6. satisfies all possibilities since if is positive we use and if is negative we use
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7. Thus, we get the solution for We generally change this equation by multiplying both sides by , then we simplify and get
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8. We generally skip all the intermediate steps for an equation like and get
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9. For quadratic equations such as we solve and get
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10. For quadratic equations that are not expressed as an equation between two squares, we can always express them as If this equation can be factored, then it can generally be solved easily.
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11. If the equation can be put in the form then we can use the square root method described previously to solve it. The solution for this equation is
The sign of m needs to be the opposite of the sign used in
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12. Fortunately the answer is yes!
The question becomes: “Can we change the equation from the form to the form ?”
Fortunately the answer is yes!
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13. The procedure for changing is as follows
The procedure for changing is as follows. First, divide by , this gives Then subtract from both sides. This gives
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14. We pause at this point to review the process of squaring a binomial
We pause at this point to review the process of squaring a binomial. We will use this procedure to help us complete the square.
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15. Recall that If we let we can solve for to get
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16. Substituting in we get Using the symmetric property of equations to reverse this equation we get
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17. Now we will return to where we left our original equation
Now we will return to where we left our original equation. If we add to both sides of we get or
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18. We can now solve this by taking the square root of both sides to get
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19. is known as the quadratic formula
is known as the quadratic formula. It is used to solve any quadratic equation in one variable. We will show how the quadratic equation is used in the example that follows.
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20. First, we start with an equation Then we change it to From this we get
Remember
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21. We then substitute into the quadratic formula, simplify and get our values for .
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22. Doing so we get
Remember the quadratic equation is and the values are a = 3, b = 11, c = -20
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23. This gives us two values for , and
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24. The equation can be factored into which will give us the same solutions as the quadratic formula. However, the beauty of using the quadratic formula is that it works for ALL quadratic equations, even those not factorable (and even when is negative).
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25. To review—the steps in using the quadratic formula are as follows:
Set the equation equal to zero, being careful not to make an error in signs.
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26. 2. Determine the values of a, b, and c after the equation is set to zero.
Remember
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27. 3. Substitute the values of a, b, and c into the quadratic formula.
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28. 4. Simplify the formula after substituting and find the solutions.
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29. Contact Us At: The MEnTe Program / Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA Phone: (323) Fax: (323) Us At: Our Websites:
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