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To find the value m such that the two roots are equal.
We know that whem the two roots are equal , the left side expression 4x^2+44x+m should be a perfect square.
Therefore 4x^2+44x+m = (X+Y)^2 = X^2+2XY +Y^2.
Therefore identifying both sides term by term,
X^2 = 4x^2 . So X = 2x.
Y^2 = m . So Y = sqrt(m)
2XY = 2(2x)*sqrtm = 44x
4x*sqrtm = 44x
sqrtm = 44x/4x =11
m = 11^2 = 121
So m = 121
We are given the equation 4x^2+44x+m=0.
We'll use the relation that for an equation ax^2 + bx +c = 0
The roots are given by : [-b +sqrt(b^2-4ac)] / 2a and [-b - sqrt(b^2-4ac)]
Now as the roots are equal it implies that sqrt(b^2-4ac) = 0
we have a = 4 , b = 44 and c =m
substituting we get
sqrt [ 44^2 - 4*4*m] =0
=> 44^2 - 4*4*m =0
=> 44^2 = 16 m
=> m = 44^2 / 16
=> m = 121
Therefore the value of m for the equation having equal roots is 121.
[ We can see that the root of the equation 4x^2 + 44x + 121 =0 is only -5.5 ]
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