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C1 - Completing the square

1

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

17

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

OP

1

Then i get???

(x+2k)^2-c

but how does that go to

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

Reply 3

TomLeigh

2

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

12

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

2

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

10

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 $(x+k)^2$ 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

how do you find the roots then once i have done that

Reply 7

have

17

Original post

by damn daniel yeah

how do you find the roots then once i have done that

That's just GCSE. You rearrange to get x on it's own in the equation. x=....

C1 - Completing the square

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