SOLUTION: Solve by any method x2 + 6x –1 = 0

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Question 280212: Solve by any method

x2 + 6x –1 = 0
Found 2 solutions by jim_thompson5910, richwmiller:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E2%2B6x-1=0 Start with the given equation.


Notice that the quadratic x%5E2%2B6x-1 is in the form of Ax%5E2%2BBx%2BC where A=1, B=6, and C=-1


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-%286%29+%2B-+sqrt%28+%286%29%5E2-4%281%29%28-1%29+%29%29%2F%282%281%29%29 Plug in A=1, B=6, and C=-1


x+=+%28-6+%2B-+sqrt%28+36-4%281%29%28-1%29+%29%29%2F%282%281%29%29 Square 6 to get 36.


x+=+%28-6+%2B-+sqrt%28+36--4+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-1%29 to get -4


x+=+%28-6+%2B-+sqrt%28+36%2B4+%29%29%2F%282%281%29%29 Rewrite sqrt%2836--4%29 as sqrt%2836%2B4%29


x+=+%28-6+%2B-+sqrt%28+40+%29%29%2F%282%281%29%29 Add 36 to 4 to get 40


x+=+%28-6+%2B-+sqrt%28+40+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%28-6+%2B-+2%2Asqrt%2810%29%29%2F%282%29 Simplify the square root (note: If you need help with simplifying square roots, check out this solver)


x+=+%28-6%29%2F%282%29+%2B-+%282%2Asqrt%2810%29%29%2F%282%29 Break up the fraction.


x+=+-3+%2B-+sqrt%2810%29 Reduce.


x+=+-3%2Bsqrt%2810%29 or x+=+-3-sqrt%2810%29 Break up the expression.


So the solutions are x+=+-3%2Bsqrt%2810%29 or x+=+-3-sqrt%2810%29

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B6x%2B-1+=+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=%286%29%5E2-4%2A1%2A-1=40.

Discriminant d=40 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-6%2B-sqrt%28+40+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%286%29%2Bsqrt%28+40+%29%29%2F2%5C1+=+0.16227766016838
x%5B2%5D+=+%28-%286%29-sqrt%28+40+%29%29%2F2%5C1+=+-6.16227766016838

Quadratic expression 1x%5E2%2B6x%2B-1 can be factored:
1x%5E2%2B6x%2B-1+=+1%28x-0.16227766016838%29%2A%28x--6.16227766016838%29
Again, the answer is: 0.16227766016838, -6.16227766016838. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B6%2Ax%2B-1+%29