If #8sqrt(4z^2 - 43) = 40#, what is the value of #z#?

Answer 1

Assumption: the question is: #" "8sqrt(4z^2)-43=40#

#z=+83/16 ->5 3/16#

#color(brown)("The objective is to manipulate the equation such that you have a")##color(brown)("single z. This is to be on one side of the equals sign and everything")# #color(brown)("else on the other side.")#

This is done in stages. First you have all the terms with z on the LHS of = and all the terms without z on the other side. Then you manipulate the LHS side until the only thing left is the single z.

#color(blue)("Step 1 - Isolate "sqrt(4z^2)#

Add 43 to both sides giving

#8sqrt(4z^2)+0=40+43#
Divide both sides by 8. Same as multiply by #1/8# to get rif og the 8 from #8sqrt(4z^2)#
#8/8xxsqrt(4z^2)=83/8#
But #8/8=1#
#sqrt(4z^2)=83/8# '~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Step 2 - Isolate z")#
But #4 = 2^2 -> 4z^2 = 2^2z^2#
Write as:#" "sqrt(2^2z^2)=83/8#

Taking the root

#=>+-2z=83/8#
divide both side by 2 to get rid of the 2 from #2z#
#+-z=83/16-> 5 3/16#
After testing #z=+83/16#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Check: Consider the left hand side only for #z=+83/16#
#8sqrt(4z^2)-43#
#8xx(2z)-43#
#8xx2xx83/16-43#
But #2xx8=16#
#cancel(16)xx83/(cancel(16))-43#
#83-43#
#40#

Thus LHS=RHS so

#color(red)(" The equation will not works for "z=-83/16)#
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Answer 2

#z = +-sqrt(17)#

#8sqrt(4z^2 - 43) = 40#

Isolate the square-root.

#sqrt(4z^2 - 43) = 5#
#(sqrt(4z^2 - 43))^2 = 5^2#
#4z^2 - 43 = 25#
#4z^2 = 25 + 43#
#4z^2 = 68#
#z^2 = 17#
#z = +-sqrt(17#

Checking in the original equation, both solutions work.

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Answer 3

To solve for ( z ):

[ 8\sqrt{4z^2 - 43} = 40 ]

[ \sqrt{4z^2 - 43} = 5 ]

[ 4z^2 - 43 = 5^2 ]

[ 4z^2 - 43 = 25 ]

[ 4z^2 = 25 + 43 ]

[ 4z^2 = 68 ]

[ z^2 = \frac{68}{4} ]

[ z^2 = 17 ]

[ z = \pm \sqrt{17} ]

So, the possible values of ( z ) are ( z = \sqrt{17} ) and ( z = -\sqrt{17} ).

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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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