"Find x intercept by solving the square" @Mathsportal.org

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Finding the x intercepts, the below answer at the below link, step 7 in the first table.
I have been shown this way, and shown another way using the RHS with more manipulation.

Is this the easiest way?
Is this the only answer we can ever arrive at no matter how we calculate?


x=−2/3±(√−43/18)
that is,
x1,x2=−23±1/6 √(86i)

And where did the 1/'6th come from.
www.mathportal.org/calculators/solving-equations/quadratic-equation-solver.php?val1=6&combo1=1&val2=8&combo2=1&val3=17&rb1=s

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Is this the easiest way?

for some problems yes, for others the quadratic formula is easier.

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Is this the only answer we can ever arrive at no matter how we calculate?

yes, the roots of a polynomial are what they are and any method you use to find them will find the same roots.

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And where did the 1/'6th come from.

sqrt(-43/18) = sqrt(-86/36) = 1/6 sqrt(-86) = (1/6) sqrt(86) i

so the 1/6 is the square of 86 by 36?
Also can't we surd the 86? If yes, how would that work exactly given the 1/6 in front of the 86?

no.

all they did was factor out sqrt(1/36) = 1/6

they doubled the -43 to -86 so they could turn 18 into 36 which is a perfect square. Then they factored the square root of that perfect square out.

Stu:

so the 1/6 is the square of 86 by 36?
Also can't we surd the 86? If yes, how would that work exactly given the 1/6 in front of the 86?

ROM'S ANSWER : sqrt(-43/18) = sqrt(-86/36) = 1/6 sqrt(-86) = (1/6) sqrt(86) i

I will just elaborate a bit.

sqrt(-43/18) = sqrt(-86/36) = sqrt(-86) / sqrt(36) = sqrt(-86) * 1/sqrt(36) = (1/sqrt(36)) * sqrt(-86) =

(1/6) sqrt(-86) = (1/6) sqrt(86) sqrt(-1) = 1/6 sqrt(86) i

NOTE: 1/6 is 1/sqrt (36)

In order to simplify something under a square root sign you have to be able to factor out a square number.

Here are the square numbers smaller than 86 ==> 1,4,9,16,25,36,49,64,81,
1 is not helpful and none of the others go into 86 therefore sqrt(86) cannot be simplified.
It can be estimated with a calculator but then it is not exact anymore - so we don't do that.

Have a think about it Stu.

I did mean to write 1/6 the simplified verion of the denominator, or at least to imply that. thanks.

Had classes today. What a ride.

To come!