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

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Question 149608: What three techniques can be used to solve quadratic equations? Demonstrate these techniques on the equation 12x^2-10x-42=0.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Technique #1 Factoring:

First let's factor 12x%5E2-10x-42


12x%5E2-10x-42 Start with the given expression


2%286x%5E2-5x-21%29 Factor out the GCF 2


Now let's focus on the inner expression 6x%5E2-5x-21

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Looking at the expression 6x%5E2-5x-21, we can see that the first coefficient is 6, the second coefficient is -5, and the last term is -21.


Now multiply the first coefficient 6 by the last term -21 to get %286%29%28-21%29=-126.


Now the question is: what two whole numbers multiply to -126 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of -126 (the previous product).


Factors of -126:
1,2,3,6,7,9,14,18,21,42,63,126
-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -126.
1*(-126)
2*(-63)
3*(-42)
6*(-21)
7*(-18)
9*(-14)
(-1)*(126)
(-2)*(63)
(-3)*(42)
(-6)*(21)
(-7)*(18)
(-9)*(14)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
1-1261+(-126)=-125
2-632+(-63)=-61
3-423+(-42)=-39
6-216+(-21)=-15
7-187+(-18)=-11
9-149+(-14)=-5
-1126-1+126=125
-263-2+63=61
-342-3+42=39
-621-6+21=15
-718-7+18=11
-914-9+14=5



From the table, we can see that the two numbers 9 and -14 add to -5 (the middle coefficient).


So the two numbers 9 and -14 both multiply to -126 and add to -5


Now replace the middle term -5x with 9x-14x. Remember, 9 and -14 add to -5. So this shows us that 9x-14x=-5x.


6x%5E2%2Bhighlight%289x-14x%29-21 Replace the second term -5x with 9x-14x.


%286x%5E2%2B9x%29%2B%28-14x-21%29 Group the terms into two pairs.


3x%282x%2B3%29%2B%28-14x-21%29 Factor out the GCF 3x from the first group.


3x%282x%2B3%29-7%282x%2B3%29 Factor out 7 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%283x-7%29%282x%2B3%29 Combine like terms. Or factor out the common term 2x%2B3


So 12x%5E2-10x-42 factors to 2%283x-7%29%282x%2B3%29


2%283x-7%29%282x%2B3%29=0 Set the factored expression equal to zero


Now set each factor equal to zero:

3x-7=0 or 2x%2B3=0

x=7%2F3 or x=-3%2F2 Now solve for x in each case


So our answers are

x=7%2F3 or x=-3%2F2







Technique #2 Quadratic Formula:



12x%5E2-10x-42=0 Start with the given equation.


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-10%29+%2B-+sqrt%28+%28-10%29%5E2-4%2812%29%28-42%29+%29%29%2F%282%2812%29%29 Plug in a=12, b=-10, and c=-42


x+=+%2810+%2B-+sqrt%28+%28-10%29%5E2-4%2812%29%28-42%29+%29%29%2F%282%2812%29%29 Negate -10 to get 10.


x+=+%2810+%2B-+sqrt%28+100-4%2812%29%28-42%29+%29%29%2F%282%2812%29%29 Square -10 to get 100.


x+=+%2810+%2B-+sqrt%28+100--2016+%29%29%2F%282%2812%29%29 Multiply 4%2812%29%28-42%29 to get -2016


x+=+%2810+%2B-+sqrt%28+100%2B2016+%29%29%2F%282%2812%29%29 Rewrite sqrt%28100--2016%29 as sqrt%28100%2B2016%29


x+=+%2810+%2B-+sqrt%28+2116+%29%29%2F%282%2812%29%29 Add 100 to 2016 to get 2116


x+=+%2810+%2B-+sqrt%28+2116+%29%29%2F%2824%29 Multiply 2 and 12 to get 24.


x+=+%2810+%2B-+46%29%2F%2824%29 Take the square root of 2116 to get 46.


x+=+%2810+%2B+46%29%2F%2824%29 or x+=+%2810+-+46%29%2F%2824%29 Break up the expression.


x+=+%2856%29%2F%2824%29 or x+=++%28-36%29%2F%2824%29 Combine like terms.


x+=+7%2F3 or x+=+-3%2F2 Simplify.


So our answers are x+=+7%2F3 or x+=+-3%2F2




Technique # 3 Completing the square


12+x%5E2%2B10+x-42 Start with the given expression


12%28x%5E2%2B%285%2F6%29x-7%2F2%29 Factor out the leading coefficient 12


Take half of the x coefficient 5%2F6 to get 5%2F12 (ie %281%2F2%29%285%2F6%29=5%2F12).

Now square 5%2F12 to get 25%2F144 (ie %285%2F12%29%5E2=%285%2F12%29%285%2F12%29=25%2F144)




12%28x%5E2%2B%285%2F6%29x%2B25%2F144-25%2F144-7%2F2%29 Now add and subtract this value inside the parenthesis. Notice how 25%2F144-25%2F144=0. Since we're adding 0, we're not changing the equation.



12%28%28x%2B5%2F12%29%5E2-25%2F144-7%2F2%29 Now factor x%5E2%2B%285%2F6%29x%2B25%2F144 to get %28x%2B5%2F12%29%5E2


12%28%28x%2B5%2F12%29%5E2-529%2F144%29 Combine like terms


12%28x%2B5%2F12%29%5E2%2B12%28-529%2F144%29 Distribute


12%28x%2B5%2F12%29%5E2-529%2F12 Multiply



So after completing the square, 12x%5E2%2B10x-42 becomes 12%28x%2B5%2F12%29%5E2-529%2F12.


So 12x%5E2%2B10x-42=0 is equivalent to 12%28x%2B5%2F12%29%5E2-529%2F12=0


12%28x%2B5%2F12%29%5E2-529%2F12=0 Start with completed square equation.



12%28x%2B5%2F12%29%5E2=529%2F12 Add 529%2F12 to both sides.


%28x%2B5%2F12%29%5E2=529%2F144 Divide both sides by 12.


x%2B5%2F12=0%2B-sqrt%28529%2F144%29 Take the square root of both sides.


x%2B5%2F12=sqrt%28529%2F144%29 or x%2B5%2F12=-sqrt%28529%2F144%29 Break up the expression


x%2B5%2F12=23%2F12 or x%2B5%2F12=-23%2F12 Take the square root of 529%2F144 to get 23%2F12



x=23%2F12-5%2F12 or x=-23%2F12-5%2F12 Subtract 5%2F12 from both sides.


x=3%2F2 or x=-7%2F3 Combine like terms and simplify.


So the answers are x=3%2F2 or x=-7%2F3






Technique # 4 Graphing

Simply graph y=12+x%5E2%2B10+x-42 to get


+graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C0%2C+12x%5E2-10x-42%29+ Graph of y=12+x%5E2%2B10+x-42


Now use the calculator's zero function to find the zeros at x=2.333 and x=-1.5