C1 - Completing the square

How do i do this???

a) Prove by completing the square, that the roots of the equation x^2+2kx+c=0 where k and c are constants, are -k +/- ,/(k^2-c)

The equation x^2 +2kx +81 = 0 has equal roots.

b) find the possible values of k

,/ denotes a square root

Reply 1

nuodai

Do you know how to complete the square? That is, for example, turning

x^2 + 4x + 1

into

(x + 2)^2 - 3

. This is how to do part a.

As for part b, as the equation has equal roots, it's of the form

(x - \alpha)^2

, so the discriminant must be equal to zero, i.e.

b^2 - 4ac = 0

. Obviously you need to do part a to find the value of

b = 2k

.

If you don't know how to do any of the above let me know.

Reply 2

tgr141291

Then i get???

(x+2k)^2-c

but how does that go to

-k +/- ,/(k^2-c)

Reply 3

TomLeigh

Remember when completing the square that yes you half the coefficient of x which is in this case 2k. Therefore inside the brackets you'll have (x+k)^2

If you were to then multiply this out you would get x^2 + 2k but also +k^2 which you wouldn't want. You must then subtract this away and add on the remaining c.

This then becomes (x+k)^2 - k^2 + c

Reply 4

chr15chr15

tgr141291

Then i get???

(x+2k)^2-c

but how does that go to

-k +/- ,/(k^2-c)

you should have:

(x + 2k)^2 + c - 4k^2 = 0

Reply 5

TomLeigh

chr15chr15

you should have:

(x + 2k)^2 + c - 4k^2 = 0

Wrong. His original equation is

x^2 + 2kx + c

To complete the square he will want to half the coefficient of x, (which is in this case 2k) so half of that is just simply k

This means his bracket will look like

(x+k)^2

which when opened up will give

x^2 + 2kx + k^2

This means his final equation will be

(x+k)^2 - k^2 + c

The next step is to make this

(x+k)^2 - k^2 + c = 0

and start trying to find the roots to match that in the question. Shout if you get stuck

Reply 6

damn daniel yeah

Original post by TomLeigh

Wrong. His original equation is

x^2 + 2kx + c

To complete the square he will want to half the coefficient of x, (which is in this case 2k) so half of that is just simply k

This means his bracket will look like