# 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".
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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
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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 (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=529 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 2.33333333333333, -1.5. Here's your graph:

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.