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

Algebra.Com
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)   (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".
----------------------
3 methods are:
Factoring
Completing the square
Quadratic formula
---------------------
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.
--------------
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
------------
2x + 3 = 0
x = -3/2
---------
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.
-------------
BTW, a 4th method is to use Excel, or manual methods, to graph the function and find where it crosses the x-axis.



RELATED QUESTIONS

What three techniques can be used to solve a quadratic equation? Demonstrate these... (answered by Nate)
What three techniques can be used to solve quadratic equations? Demonstrate these... (answered by jim_thompson5910)
Can somebody please help me? 2. What three techniques can be used to solve a... (answered by jim_thompson5910)
What three techniques can be used to solve a quadratic equation? Demonstrate these... (answered by RAY100)
What three techniques can be used to solve a quadratic equation? Please could you... (answered by checkley77)
please help me answer this question. What three techniques can be used to solve a... (answered by Alan3354)
What three techniques can be used to solve a quadratic equation? Demonstrate these... (answered by stanbon)
What three techniques can be used to solve a quadratic equation? Demonstrate these... (answered by stanbon)
5b. What three techniques can be used to solve a quadratic equation? Demonstrate... (answered by Alan3354)