SOLUTION: 1) Using the quadratic equation x2 - 6x + 8 = 0, perform the following tasks: a) Solve by factoring. Answer: Show work in this space. b) Solve by using the quadratic

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: 1) Using the quadratic equation x2 - 6x + 8 = 0, perform the following tasks: a) Solve by factoring. Answer: Show work in this space. b) Solve by using the quadratic      Log On


   

Question 76350: 1) Using the quadratic equation x2 - 6x + 8 = 0, perform the following tasks:
a) Solve by factoring.
Answer:
Show work in this space.

b) Solve by using the quadratic formula.
Answer:
Show work in this space.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a)
Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)
In order to factor 1%2Ax%5E2%2B-6%2Ax%2B8, first we need to ask ourselves: What two numbers multiply to 8 and add to -6? Lets find out by listing all of the possible factors of 8


Factors:

1,2,4,8,

-1,-2,-4,-8,List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to 8.

1*8=8

2*4=8

(-1)*(-8)=8

(-2)*(-4)=8

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6

||||
First Number|Second Number|Sum
1|8|1+8=9
2|4|2+4=6
-1|-8|-1+(-8)=-9
-2|-4|-2+(-4)=-6
We can see from the table that -2 and -4 add to -6.So the two numbers that multiply to 8 and add to -6 are: -2 and -4 Now we substitute these numbers into a and b of the general equation of a product of linear factors which is: %28x%2Ba%29%28x%2Bb%29substitute a=-2 and b=-4 So the equation becomes: (x-2)(x-4) Notice that if we foil (x-2)(x-4) we get the quadratic 1%2Ax%5E2%2B-6%2Ax%2B8 again

So in other words, x%5E2-6x%2B8=0 factors to %28x-2%29%28x-4%29=0
Now set each factor equal to zero:
x-2=0
x=2 Solve for x
x-4=0
x=4 Solve for x
So our solution is x=2 and x=4




b)
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-6x%2B8+=+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=%28-6%29%5E2-4%2A1%2A8=4.

Discriminant d=4 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+4+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+4+%29%29%2F2%5C1+=+4
x%5B2%5D+=+%28-%28-6%29-sqrt%28+4+%29%29%2F2%5C1+=+2

Quadratic expression 1x%5E2%2B-6x%2B8 can be factored:
1x%5E2%2B-6x%2B8+=+1%28x-4%29%2A%28x-2%29
Again, the answer is: 4, 2. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-6%2Ax%2B8+%29

So our solution is x=2, and x=4.
Notice how we got the same answer but took another route to get there.