Express 3x^2+12x+7 in the form 3(x+a)^2+b?

Question

i) Express 3x^2+12x+7 in the form 3(x+a)^2+b

ii) Hence write down the equation of the line of symetry of the curve y=3x^2+12x+7

also can you show me how you worked it out please!
all help greatly appreciated thanks!

Answer 1

You need to complete the square to get in the form you want. First, you need to factor out a 3 of the x-terms because the general form you have has a factor of 3:

3(x^2 + 4x) + 7

Looking at the general form of a squared prdouct (x+a)^2, we have:

(x+a)^2 = x^2+2ax+a^2

If we compare this expression to the one above within the parenthesis, you can deduce that 2a = 4, thus a = 2. Therefore, our squared term is of the form (x+2)^2. We know that:

(x+2)^2 = x^2+4x+4. Thus, in our original factoring, we see that we are missing a '4' term. To create this value, we need to re-write the constant values in a different way. In order to introduce a 4 term inside the parenthesis, we actually need to create a '12' term because of the factor of 3.

Thus, we can rewrite the constant term as:

7 = -5 + 12. Therefore, we have:

3(x^2 + 4x) + 7 -----> 3(x^2+4x+4) - 5

= 3(x+2)^2 - 5

Therefore, we have a = 2 and b = -5

To check if our answer is correct, multiply out our answer and see if we get the original.

3(x+2)^2-5 = 3(x^2+4x+4)-5 = 3x^2+12x+12-5 = 3x^2+12x+7

Therefore, our answer is correct
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A parabola has the general form of:

y = (x-h)^2 + k

where the parabola is symmetric when x-h=0

Using this, we see that 3(x+2)^2 - 5 is symmetric when x+2=0

Therefore, symmetry is defined at the line x = -2

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Hope this helps

Answer 2

i) 3x^2 + 12x + 7 = 3(x^2 + 4x + 4) - 3*4 +7 =
= 3 (x + 2)^2 -12 +7 = 3(x+2)^2 -5;

So we have the parabolic curve y = 3(x+2)^2 -5;
When x = -2, we have y = -5.
Any other x makes y to get positive additive 3(x+2)^2, so the curve has its minimum at x = -2.
And the parabolic curve is symmetric for x deviating to positive and to negative values from x = -2. That is why the equation
x = -2
is the equation for the line of symmetry for the curve.

Answer 3

3x²+12x+7 = 3(x² + 4x) + 7

Need to complete the square inside the brackets... knowing that (x+a)² = x²+2ax+a², the coefficient for x is 2a=4 so a=2 and therefor we need to bring the constant term a²=4 between the brackets.

... = 3(x² + 4x) + 12 - 12 + 7
= 3(x² + 4x + 4) - 5
= 3(x+2)² - 5.

The line of symmetry is at the x that will minimize the function, that is (x+2)²=0 or x = -2.

Answer 4

Complete the Square for part i

3x^2 + 12x +7

taking 3 out of the sum

3(x^2+4x+7/3)

Ignore the 3 for now

x^2+4x+7/3 = (x+2)^2 - 4 +7/3 = (x+2)^2 - 5/3

Re-introduce the 3

3((x+2)^2-5/3) = 3(x+2)^2 - 5

a=2, b=-5

ii) Not sure, try plotting the graph

Answer 5

i) 3x^2+12x+7
= 3[x^2 + 4x + (7/3)]
= 3[x^2 + 4x + (2 1/3)]
= 3[(x+2)^2 - 1 2/3]
= 3[(x+2)^2 - 5/3]
= 3(x+2)^2 - 5

ii) The vertex (or minimum turning point) is (-2, -5).
Hence the line of symmetry is y = -5.

Answer 6

i)3x^2+12x+7=3(x+a)^2+b
put x=0,
7=3a^2+b...(1)
put x=1,
22=3(1+a)^2+b
22=3a^2+6a+3+b
19=3a^2+6a +b....(2)
subtract (1) from (2)
12=6a
>>>a=2,b= -5

therefore,

3x^2+12x+7=3(x+2)^2-5

ii)when 3(x+2)^2=0,
3(x+2)^2-5 = -5 is a turning point
and is a minimum since 3(x+2)^2
cannot be -ve
3(x+2)^2=0 when x+2=0
ie x= -2
hence, the line of symmetry of
3x^2+12x+7 is x = -2

i hope that this helps

Answer 7

expanding 3(x+a)^2+ b, we get:
= 3(x^2+6ax+a^2)+b
= 3x^2 + 18ax + 3a^2 + b
which means that 18ax = 12x therefore, a = 2/3
which also means that 3a^2 + b = 7 i.e. 3(2/3)^2 + b = 7
therefore, b = 7 - (3(2/3)^2 = 7 - (4/3) = 5 2/3 or 5.667
so,substituting back, we have:
3(x+(2/3))^2 + 5.667