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If x is real, prove that xx25x+9 lies between 111 and 1.

Answer
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Hint: Here in this question it is given that x is real and also given an equation in x and you have to prove that range of x is between 111 and 1. In order to do so, you have to make a quadratic equation by considering it equal to another variable and then apply the discriminant conditions for having real roots.

Complete step by step solution:It is the question that is given an equation in x.
Let us consider xx25x+9=y
Now on cross multiplication, we get
x=yx2(5y)x+9y
On making the above equation a quadratic in terms of x, we get
yx2(1+5y)x+9y=0 ……….(i)
Here in the question, it is also given that the x is real.
Now according to the question, this quadratic equation must have real roots. In doing so, the discriminant for this quadratic equation must be greater than or equal to zero.
Discriminant must be 0
b24ac0 ……….(ii)
In order to find a,b and c, compare equation (i) with the standard quadratic equation ax2+bx+c=0
On comparing, we get
a=y , b=(1+5y) and c=9y
Now putting all that values in equation (ii), we get
(15y)24×y×9y0
1+25y2+10y36y20
On further solving, we get
1+10y11y20
Now, solving this equation using factorisation method
1y+11y11y20
On further making factors, we get
(1y)(11y2+1)0
Now on taking minus sign common, the inequality symbol gets reversed
(11y+1)(y1)0
On solving this, we get range as
y[111,1]
Hence, it is proved that the xx25x+9 lies between 111 and 1

Note:
The discriminant is the part of the quadratic formula underneath the square root symbol b24ac . The discriminant tells us whether there are two solutions, one solution, or no solutions.
The discriminant can be positive, zero, or negative, and this determines how many solutions there are to the given quadratic equation.
-A positive discriminant indicates that the quadratic has two distinct real number solutions.
-A discriminant of zero indicates that the quadratic has a repeated real number solution.
-A negative discriminant indicates that neither of the solutions is real numbers.