Square root of x^2-5x+4 is defined for x = ?
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The square root of x^2 - 5x + 4 is a real value if x^2 - 5x + 4 >=0
x^2 - 5x + 4 = 0
=> x^2 - 4x - x + 4 =0
=> x(x - 4) - 1(x - 4) = 0
=> (x - 1)(x - 4) = 0
The roots are x = 1 and x = 4
For values of x such that 1< x < 4, x^2 - 5x + 4 < 0
Therefore the square root is defined when
x <= 1 and x >= 4.
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The square root exists if and only if only the expression x^2-5x+4 is positive or zero.
To determine the range of x values for the expression x^2-5x+4 to be positive, we'll have to calculate the roots of the expression x^2-5x+4.
x1=[-(-5)+sqrt(25-16)]/2
x1=(5+3)/2
x1=4
x2=[-(-5)-sqrt(25-16)]/2
x2=(5-3)/2
x2=1
The radicand is positive if x belongs to the ranges (-infinite,1] or [4,+infinite) and it is negative for (1,4).
So, the square root is defined if x belongs to the reunion of intervals: (-infinite,1] U [4,+infinite).
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