How do you solve #2^ { 3} = \frac { x ^ { 2} } { 6- x }#?

Answer 1

See a solution process below:

First, evaluate the term on the left side of the equation:

#2^3 = x^2/(6 - x)#
#8 = x^2/(6 - x)#
Next, multiply each side of the equation by #color(red)((6 - x))# to eliminate the fraction while keeping the equation balanced:
#8color(red)((6 - x)) = x^2/(6 - x) xx color(red)((6 - x))#
#(8 xx color(red)(6)) - (8 xx color(red)(x)) = x^2/color(red)(cancel(color(black)(6 - x))) xx cancel(color(red)((6 - x)))#
#48 - 8x = x^2#
Now, subtract #color(red)(48)# and add #color(blue)(8x)# to each side of the equation to put the equation in standard form while keeping the equation balanced:
#48 - color(red)(48) - 8x + color(blue)(8x) = x^2 + color(blue)(8x) - color(red)(48)#
#0 -0 = x^2 + 8x - 48#
#0 = x^2 + 8x - 48#
#x^2 + 8x - 48 = 0#

Next, factor the left side of the equation as:

#(x + 12)(x - 4) = 0

Now, solve each term on the left for #0#

Solution 1:

#x + 12 = 0#
#x + 12 - color(red)(12) = 0 - color(red)(12)#
#x + 0 = -12#
#x = -12#

Solution 2:

#x - 4 = 0#
#x - 4 + color(red)(4) = 0 + color(red)(4)#
#x - 0 = 4#
#x = 4#
The Solution Is: #x = {-12, 4}#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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