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- 5 years agoLemme break it down for you :) I'm in Year 7 so if it is not a very good explanation I'm sorry :)

x^2 - 4x + c has to have one solution only.

An quad which has only one solution is written in the form x^2 - ax + a^2/4 = 0 to factorise to:

(a - a/2)^2 = 0,giving a solution of:

x = a/2.

Now, how did I get that a^2/4 ? ? ?

Consider the following:

What do I have to add to x^2 - 6x such that the result is a perfect square, that is, it factorises to (x + ?)^2 ? ? ?

You can add a variety of numbers to x^2 - 6x such that it factorises:

x^2 - 6x + 5 = (x - 5)(x - 1)

x^2 - 6x + 8 = (x - 2)(x - 4).

But, if you add 9:

x^2 - 6x + 9 = (x - 3)^2.

Similarly:

x^2 - 8x + 16 = (x - 4)^2

x^2 - 2x + 1 = (x - 1)^2

x^2 - 12x + 36 = (x - 6 )^"

x^2 - 14x + 49 = (x - 7 )^2

x^2 - 18x + 81 = (x - 9)^2

Do you notice any relationship between the coefficient of x, and the constant term in each one ? That is, can you think of any thing that you can do to the number of x to get the constant term:

What do you do to -8 to get 16, that you do to -2 to get 1, that you do to -12 to get 36, -14 to get 49, -18 to get 81 ?

-8........16

-2.........1

-12........36

-14.........49

-18.........81 ? ? ? ? ? ? ?

You should be seeing that if you take the coefficient of x, half it and then square it, you get the final number:

1/2 of -8 = -4, and (-4)^2 = 16

1/2 of -2 = -1, and (-1)^2 = 1

1/2 of -12 = -6, and (-6)^2 = 36

1/2 of -14 = -7, and (-7)^2 = 49

1/2 of -18 = -9,and (-9)^2 = 81.

So, in order to write x^2 + ax + ? as a perfect square, that is as (x + *)^2, half the coefficient of x, square it, and add it.

So:

What is the value of c such that x^2 + 4x + c has only one solution ?

Take the coefficient of x which is 4, half it to give 2, square it to give 4, and add:

x^2 + 4x + 4 = 0, so

(x + 2)^2 = 0, so

c = 4.

I hope you can follow this rather long winded explanation. It is important, and it is called the method of completing the square. I really do hope this has helped :D5 years agoAll the colons next to opening brackets have been replaced with these: :( so sorry :'(

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