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 Depdendent Variable

 Number of equations to solve: 23456789
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 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

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HOW TO SOLVE ANY QUADRATIC EQUATION: First of all, make sure that you know the
that is second degree. The Quadratic Formula is the formula used to solve any quadratic equation. You
have probably heard of this formula, and some of you have probably already used it. Actually any quadratic
equation can be solved by completing the square, but the Quadratic Formula gives us a direct way to get
the solutions that is often quicker.

WHERE DOES IT COME FROM?: You are certainly NOT responsible for deriving the Quadratic
Formula. I show the derivation here for those that are interested and so that you are at least aware that it
is a direct consequence of Solving by Completing the Square, which in turn was dependent upon Extracting
Roots. First: The equation must be set equal to 0. Start with a general second degree equation equal to zero. Divide every term by a to get coefficient of x^2 to be 1, & keep equation equivalent. Move constant to RIGHT & set up blanks for Completing the Square.
Now figure out what has to be added:  LEFT side is now a Perfect Square Trinomial LEFT - Binomial Squared; RIGHT - Common denominator: Extract Roots after combining RIGHT to one fraction. “Rearrange” numerator terms & simplify square root of denominator. Isolate x Add fractions together, using common denominator, 2a.

IF ax^2 + bx + c = 0 THEN: Gives the solutions for x to the above equation.

REMEMBER: The equation MUST start set equal to 0 before using the formula!
ALSO: Notice the position of the fraction bar under the entire numerator. Please be NEAT & CAREFUL

DO I HAVE TO MEMORIZE IT?: YES! Rather, I hope you will LEARN it from practicing.
But that is the BEST way to learn it is to PRACTICE, PRACTICE, PRACTICE, · · ·. Furthermore, just
“substituting” the correct numbers into the correct formula is only about HALF the problem. You must
know how to correctly SIMPLIFY what you get.

EXAMPLE: SOLVE using the Quadratic Formula: FIRST the equation MUST be set equal to 0 before the Formula can be applied: NOW determine: a = 3, b = −7, c = −4. Watch your signs. To learn quickly, WRITE and SAY the Formula each time. SHOW your substitution step. Proceed with the Arithmetic. This one required little simplification.

WARNING: Notice in Step 1, that the correct substitution for b2 = (−7)2 NOT −72. Remember our
discussion in class as to Order of Operations: −72 = −49. If you wirte −72 and then 49 in the next
step, you have made two errors!

NEXT EXAMPLE: SOLVE using the Quadratic Formula: IS set equal to 0 ready for the Formula, with a = 3, b = −4, c = 4. WRITE and SAY the Formula. SHOW your substitution step. Proceed with the Arithmetic.  “Take i out” Simplify Radical (You can combine last two steps easily.) Factor a 2 out of numerator to cancel with denominator: So you can see on a problem like this that the substitution was just the beginning!

WARNING: One of the MOST common MISTAKES is “cancelling” an “added term” from the numerator
in one of the last steps. This would be a VSE - Very Serious Error - DON’T DO IT!

NOTE: I have NO problem with you combining some steps. I tried to put in everything in these examples.
I will ask you to SHOW your SUBSTITUTION STEP, so that I know that you are doing this correctly,
and to discern where possible errors are coming from. After that you may combine some steps, but on a
problem as long as this last one, I would doubt that anyone would do all the steps in their head, so put down
the steps that you need, rather than working it on scratch paper and just giving me an answer - UNacceptable!

LAST EXAMPLE: SOLVE using the Quadratic Formula:

1. x2 + 5x + 6 = 0 IS set equal to 0 ready for the Formula, with a = 1, b = 5, c = 6. WRITE and SAY the Formula. SHOW your substitution step. Proceed with the Arithmetic. Now this serves to illustrate what must be done when the radical “comes out even”. It would not be
proper to leave the answer in this form because the two individual answers: simplify to two other numbers: respectively.

7. x = −2 or x = −3 Answers correctly simplified.

Actually, if you had just been asked to SOLVE this equation any way you wanted to, you would have
factored one this simple; however, I used this to illustrate that when you do use the Quadratic Formula, or
Solve by Completing the Square, or by Extracting Roots and the solution comes out with NO Radical left,
you MUST simplify the two numbers individually, rather than leave it in “± notation.”