Completing the square with rational numbers?

Question

Can someone please explain this process to me. I'm a distance student and really need some help!

For example:

Express the following in the form (x+a)^2 + b where a and b are rational.

x^2 + 6x + 7

Thankyou!!

Answer 1

(x+3)^2 - 2

If you look at the x^2 + 6x part, you should ask yourself "what am i missing to complete this square?"

If I had x^2 + 6x + 9, I'd have a perfect square: (x + 3 )^2.

I have 7 instead, so how do I get from 9 to 7? -2

Combine the two parts. (x + 3 )^2 - 2

Answer 2

x^2 + 6x + 7 = (x+3)^2 - 2

because we know that
(x+3)^2 = x^2 + 3x + 9 but what we need is x^2 + 6x + 7. So that is why we have to subtract 2 from (x+3)^2.

As for why it must be x+3 and nothing else, it is because when (x+3)^2 is expanded, the x^2 and x values are the same as those in the question.
The method is to divide the x term, in this case 6, by 2.

Similarly,
x^2 - 4x + 7 = (x-2)^2 + 3
x^2 + 8x = (x+4)^2 - 64

Answer 3

Before starting, you need to know that :

(c+d)^2= c^2 + 2cd + d^2.

now, the example that you have given,

x^2 + 6x + 7

we have to reduce the equation to a form similar to (a+b)^2, so
we can also write the above equation as

x^2 + 2*x*3 + 7

if we compare x^2 + 2*x*3 to c^2 + 2cd, then we see that c=x, and d=3. so the equation can now be written as:

x^2 + 2*x*3 +9 -9+ 7, which will give the same thing. ( +9 - 9 = 0)

Now, x^2 + 2*x*3 +(3)^2 and be written as a whole square , (x+3)^2, so,

(x^2 + 2*x*3 +9) -9+ 7 is now

(x+3)^2 -9 +7,

= (x+3)^2 -2.

Hope this helps.

Plus, suppose that you are asked 3x^2 + 6x + 7, then first take the constant,3, common.

3(x^2 + 2x) + 7. You can now continue simplifying as shown above.

Answer 4

Note: (x-a)^2=x^2-2ax+a^2
x^2+6x+7
One could certainly add and subtract a number d^2 from the expression.
=[x^2+6x+d^2]+(7-d^2)
Suppose I chose this d such that -2d=6 or d=-6/2=-3 then
=(x-(-3))^2+(7-9)
=(x+3)^2-2

This is how one "completes the square".

Answer 5

x² + 6x + 7 = 0

x² + 6x + 7 - 7 = - 7

x² + 6x + 9 = - 7 + 9

(x + 3) (x + 3) = 2

(x + 3)² = ± √2

x = - 3 ± √2

Answer 6

(x + 3)^2 - 2

Answer 7

I think its (x+3)^2 - 2.

Answer 8

Beats me!