SOLUTION: Hi! On the following problem I keep getting the incorrect answer. I think I'm close to the right answer just making a small error somewhere. Can you help me? Solve the equati

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Hi! On the following problem I keep getting the incorrect answer. I think I'm close to the right answer just making a small error somewhere. Can you help me? Solve the equati      Log On


   

Question 196635: Hi!
On the following problem I keep getting the incorrect answer. I think I'm close to the right answer just making a small error somewhere. Can you help me?
Solve the equation for the variable x. The constant a represents a positive real number.
a^3x^3 + 64 = 0
Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The constant a represents a positive real number.
a^3x^3 + 64 = 0
(ax+4)*(a^2x^2 - 4ax + 16) = 0
x = -4/a
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Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-4x%2B16+=+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=%28-4%29%5E2-4%2A1%2A16=-48.

The discriminant -48 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -48 is + or - sqrt%28+48%29+=+6.92820323027551.

The solution is , or
Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B16+%29

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Reducing gives:
2 ± isqrt(12)
Then taking the "a" into account:
x = (2 ± 2i*sqrt(3))/a


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
a%5E3x%5E3+%2B+64+=+0 Start with the given equation.


%28ax%29%5E3+%2B+64+=+0 Rewrite a%5E3x%5E3 as %28ax%29%5E3


%28ax%29%5E3+%2B+4%5E3+=+0 Rewrite 64 as %284%29%5E3


%28ax%2B4%29%28%28ax%29%5E2-%28ax%29%284%29%2B%284%29%5E2%29+=+0 Factor the left side using the sum of cubes formula


Recall, the sum of cubes formula is A%5E3%2BB%5E3=%28A%2BB%29%28A%5E2-AB%2BB%5E2%29


%28ax%2B4%29%28a%5E2x%5E2-4ax%2B16%29+=+0 Multiply and simplify


ax%2B4=0 or a%5E2x%5E2-4ax%2B16=0 Use the zero product property to break up the factors


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Let's solve the first equation: ax%2B4=0


ax%2B4=0 Start with the first equation.


ax=-4 Subtract 4 from both sides.


x=-4%2Fa Divide both sides by "a" to isolate "x".


So the first solution is x=-4%2Fa

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Now let's solve the second equation: a%5E2x%5E2-4ax%2B16=0


a%5E2x%5E2-4ax%2B16=0 Start with the second equation.


Notice we have a quadratic in the form of Ax%5E2%2BBx%2BC where A=a%5E2, B=-4a, and C=16


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


Plug in A=a%5E2, B=-4a, and C=16


x+=+%284a+%2B-+sqrt%28+%28-4a%29%5E2-4%28a%5E2%29%2816%29+%29%29%2F%282%28a%5E2%29%29 Negate -4a to get 4a


x+=+%284a+%2B-+sqrt%28+16a%5E2-4%28a%5E2%29%2816%29+%29%29%2F%282%28a%5E2%29%29 Square -4a to get 16a%5E2


x+=+%284a+%2B-+sqrt%28+16a%5E2-64a%5E2+%29%29%2F%282a%5E2%29 Multiply


x+=+%284a+%2B-+sqrt%28+-48a%5E2+%29%29%2F%282a%5E2%29 Combine like terms.


x+=+%284a+%2B-+sqrt%28+-1%2A16%2A3%2Aa%5E2+%29%29%2F%282a%5E2%29 Factor -48 into -1%2A16%2A3


Break up the square root.


x+=+%284a+%2B-+i%2A4%2Asqrt%283%29%2Aa+%29%2F%282a%5E2%29 Simplify the square roots and replace sqrt%28-1%29 with "i".


Note: i=sqrt%28-1%29


x+=+%284a+%2B-+4a%2Asqrt%283%29%2Ai+%29%2F%282a%5E2%29 Rearrange the terms.


x+=+%284a+%2B+4a%2Asqrt%283%29%2Ai+%29%2F%282a%5E2%29 or a+=+%284a+-+4a%2Asqrt%283%29%2Ai+%29%2F%282a%5E2%29 Break up the "plus/minus"


x+=+%282+%2B+2%2Asqrt%283%29%2Ai+%29%2Fa or x+=+%282+-+2%2Asqrt%283%29%2Ai+%29%2Fa Reduce


So the next two solutions are x+=+%282+%2B+2%2Asqrt%283%29%2Ai+%29%2Fa or x+=+%282+-+2%2Asqrt%283%29%2Ai+%29%2Fa


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Answer:

So the three solutions are x=-4%2Fa, x+=+%282+%2B+2%2Asqrt%283%29%2Ai+%29%2Fa or x+=+%282+-+2%2Asqrt%283%29%2Ai+%29%2Fa