SOLUTION: What three techniques can be used to solve a quadratic equation? Demonstrate these techniques on the equation "12x^2 - 10x - 42 = 0".

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Question 154113: What three techniques can be used to solve a quadratic equation? Demonstrate these techniques on the equation "12x^2 - 10x - 42 = 0".
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
What three techniques can be used to solve a quadratic equation? Demonstrate these techniques on the equation "12x^2 - 10x - 42 = 0".
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3 methods are:
Factoring
Completing the square
Quadratic formula
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12x^2 - 10x - 42 = 0
Divide by 2. Not necessary, but it makes it simpler to factor.
6x^2 - 5x - 21 = 0
Factoring is a trial and error process. If we use the quadratic equation to solve, we will know the factors.
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 6x%5E2%2B-5x%2B-21+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-5%29%5E2-4%2A6%2A-21=529.

Discriminant d=529 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--5%2B-sqrt%28+529+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+529+%29%29%2F2%5C6+=+2.33333333333333
x%5B2%5D+=+%28-%28-5%29-sqrt%28+529+%29%29%2F2%5C6+=+-1.5

Quadratic expression 6x%5E2%2B-5x%2B-21 can be factored:
6x%5E2%2B-5x%2B-21+=+%28x-2.33333333333333%29%2A%28x--1.5%29
Again, the answer is: 2.33333333333333, -1.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+6%2Ax%5E2%2B-5%2Ax%2B-21+%29

The factors shown by the on-site solver are not exactly right, it divides so the the coefficient of the x^2 is one.
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Factoring:
Using the results from above, we can multiply by 3 and 2 and get the factors.
(3x - 7)*(2x + 3) = 6x^2 - 5x - 21
So, 3x - 7 = 0
x = 7/3
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2x + 3 = 0
x = -3/2
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3rd method, Completing the square:
12x^2 - 10x - 42 = 0
Divide by 12
x^2 - (5x/6) - 7/2 = 0
The x term's coeff, -5/6, is 2 times the sqrt of the numeric term, so the NM, the last term will be (-5/12)^2, or 25/144.
x^2 - (5x/6) - 7/2 = 0
x^2 - (5x/6) = 7/2
x^2 - (5x/6) + 25/144 = 7/2 + 25/144
(x - 5/12)^2 = 7/2 + 25/144 = 504/144 + 25/144 = 529/144 = (23/12)^2
(x - 5/12)^2 = (23/12)^2
Take sqrt of both sides:
x - 5/12 = 23/12 or -23/12
x = 28/12 and x = -18/12
x = 7/3 and -3/2
Same answers, but a lot more work. That's why we do the completion of the square ONE TIME with literal terms, ax^2 + bx + c = 0, to find the quadratic equation, then NEVER use completion of squares again.
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BTW, a 4th method is to use Excel, or manual methods, to graph the function and find where it crosses the x-axis.