question (2)  solve 5y^{2} + 12y + 4 = 0

Step 
Explanation 
preresult 
result after applying step 
[0] 
writing given equation 
5y^{2} + 12y + 4 = 0 
[1] 
Substract constant term '4' from both sides of equation and simplifying 

5y^{2} + 12y + 4  4 = 0  4 5y^{2} + 12y = 4 
[2] 
Now divide the equation obtained in previous step throughout by the coefficient of y^{2} or the number in front of y^{2} ,which is 5 in this case
and simplifying 
i.e 
(5y^{2}/5) + 12y/5 = 4/5
y^{2} + (12/5)y = 4/5

[3] 
Now find half of '12/5', the coefficient of y or find half of the number in front of y of the last equation obtained in step(2)
, square the result and add it to both sides of equation 
(1/2)x(12/5) = 6/5
6^{2}/5^{2} 
y^{2} + (12/5)y + 6^{2}/5^{2}= 4/5 + (6/5)^{2}

[4] 
Since (6/5)+(6/5)= 12/5
and (6/5)X(6/5)= 6^{2}/5^{2}
we factor the left hand side of last equation obtained in step(3)
to factor all we do is take the variable which is y ,in this case, and add it to the
the constant '6/5' that is squared on the left hand side of previous equation,
and square the result 
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[ y + (6/5)]^{2}= (4/5)+(6/5)^{2} 
[5] 
Simplify right hand side of equation
since 

[ y + (6/5)]^{2}= (4/5)+(6/5)^{2}
[ y + (6/5)]^{2} = (4/5)+(36/25)
[ y + (6/5)]^{2} = (20 + 36)/25
[ y + (6/5)]^{2} = 16/25 
[6] 
Now solve the last equation obtained in step(5) for y,
by taking the square root on both sides of equation
now since the square root of any number R^{2} is that number
which when multiplied by itself gives R^{2}
note that R x R = R^{2} , and
R x R = R^{2} so, the square root of
R^{2} is R and R ,thus the square root of a number
includes the negative root also ,for example the square roots of 25 are
5 and 5 ,since 5x5=25 and 5x5=25.

Since the squareroot of a number squared is the number itself ,the squareroot [ y + (6/5)]^{2} is
y + (6/5)

y +(6/5)= (+/) Ö(16/25)

[7] 
Since Ö(16/25) = Ö16/Ö25 = (+/)4/5 ,
simplifying the right hand side of the previous equation gives:

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y+(6/5) = (+/)4/5 
[8] 
(+/) implies that 
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y+(6/5) = +(4/5) and y+(6/5)= (4/5) 
[9] 
Now solve both equations obtained in step[8] 
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y+(6/5) = 4/5
y = (4/5)(6/5)
y =  2/5

and 
y+(6/5) = 4/5
y = (4/5)(6/5)
y = 10/5
y = 2


Hence the solutions to given equation are 
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