Question

Show that whatever value of x, the value of the polynomial p(x)=x^2-2x+6 is always not less than 5.

Answer 1

Answer:

the discriminant = b² - 4.a.c

here a=1, b=-2 and c=5

So, the discriminant = (-2)² - 4.1.5 = -16

As the discriminant is negative, the quadratic equation has no real root.

And if we put x=0, then the equation will be 5 which is positive so the equation totally lies above the real axis. So the sign of the equation is same as the sign of a i.e positive.

You can prove it by another method.

x²−2x+5 = x²−2x+1+4

= (x-1)² + 4

= (x-1)² + 2²

So, we can see that the equation is the sum of two squared number which can't be negative. Thus, the equation is positive.

Another method is,

Note that the first derivative of the equation is 2x-2. So the point where the first derivative vanishes is at x=1. Then the value of the second derivative at this point is equal to 2. As 2 is positive, the function has a local minimum at x=1. As we see the least value of the function is at x=1, that is 4. So the function always has a positive value.

Hope you have found your answer. Feel free to ask more questions.

Thank you. Happy to help you.

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