# P1 Maths Question, Help.

I'd be gratefull if anyone could show me how to do this question:

f(x) = x^2 - kx + 9, where k is a constant

a) Find the set of values of k for which the equation f(x) = 0 has no real solutions.

Given that k = 4

b) Express f(x) in the form (x-p)^2 + q, where p and q are constants to be found.

c) Write down the minimum value of f(x) and the value of x for which this occurs.

I've really forgotten how to do it.
Bebop
I'd be gratefull if anyone could show me how to do this question:

f(x) = x^2 - kx + 9, where k is a constant

a) Find the set of values of k for which the equation f(x) = 0 has no real solutions.

Given that k = 4

b) Express f(x) in the form (x-p)^2 + q, where p and q are constants to be found.

c) Write down the minimum value of f(x) and the value of x for which this occurs.

I've really forgotten how to do it.

a.
(b^2-4ac) must be positive to give real solution so,
therefore b^2>=36
so b>=¦6¦ (modulus 6 i think) [to give realy] so u want -6<k<6 to give no real solution :-)
b.complete square? (x-2)^2-4+9=(x-2)^2+5
c. differentitate
d(f(x))/dx=2x-4
when 2x-4=0, x=2
Bebop
I'd be gratefull if anyone could show me how to do this question:

f(x) = x^2 - kx + 9, where k is a constant

a) Find the set of values of k for which the equation f(x) = 0 has no real solutions.

Given that k = 4

b) Express f(x) in the form (x-p)^2 + q, where p and q are constants to be found.

c) Write down the minimum value of f(x) and the value of x for which this occurs.

I've really forgotten how to do it.

Its a quadratic equation with
a = 1
b = -k
c = 9

a quadratic equation has no real root if

b^2 < 4ac

therefore, substituting gives:

(-k)^2 < 4*1*9

K^2 < 36

K < plus OR minus 6

so inequality is :

-6 < k <6

For second part use completing the square method to get:

(x - 2)^2 + 5

For the third part differentiate the equation to get:

2x - 4

equate this to zero to get:

x = 2

which gives f(x) = 5

differentiate again to get 2, which is greater than 0, therefore their is only one minimum value for the function.
Thank you both very, very much!

Well since I'm brushing up on the p1 I might as well get another thing cleared up.

In integration I've never felt confident knowing what to do when you have to integrate a term containing a real number divided by the root of x.

Eg. 3&#8730;x + 12/&#8730;x

I know that the root of x is equvalent to x to the power a half.

Anyway I'd appreciate any ideas on this.
f'(x) = 3&#8730;x + 12/&#8730;x
= 3x^1/2 + 12x^-1/2

f(x) = 2x^3/2 + 24x^1/2 + c

I think.
Bebop
Thank you both very, very much!

Well since I'm brushing up on the p1 I might as well get another thing cleared up.

In integration I've never felt confident knowing what to do when you have to integrate a term containing a real number divided by the root of x.

Eg. 3&#8730;x + 12/&#8730;x

I know that the root of x is equvalent to x to the power a half.

Anyway I'd appreciate any ideas on this.

like you said, &#8730;x is the x to the half, so just treat it as x^n with n=1/2, as you would any other integral of powers of x.

ie &#8747;&#8730;x dx = &#8747;x^1/2 dx = (x^3/2)/3/2 +C = 2/3 x&#8730;x + C
and &#8747;(dx/&#8730;x) = &#8747;x^-1/2 dx = (x^1/2)/1/2 +C = 2&#8730;x +C
Thank you, I'm extremely grateful to you all.