Hello, I am having the worst time trying to solve these parabolas and putting them into x-h = a(y-k)^2 form. :(
There are two problems that i keep doing something wrong. could someone solve them so I can have a set up for the rest of my problems? :)
3y^2+6y+108x-969=0
and
x^2+14x=44y+313=0
All help is greatly appreciated!
108x = -3y^2 - 6y + 969
108x = -3(y^2 + 2y + 1) + 972
x = -1/36 (y+1)^2 + 9
I'm not too sure how to do the second one. I'm really sorry.
Can you solve this one?:
x^2-12x-48y-372=0
& I made a mistake one the second one!!
Its actually:
x^2+14x+44y+313=0 !!
x^2+14x+44y+313=0
44y = -x^2 - 14x - 313
44y = -(x^2 + 14x +49-49 - 313
44y = -( (x+7)^2 - 49 ) - 313
44y = -(x+7)^2 + 49 - 313
44y = -(x+7)^2 - 264
44y + 264 = -(x+7)^2
44(y + 6) = -(x+7)^2
y + 6 = (-1/44) (x+7)^2 , in the form that was requested.
x^2-12x-48y-372=0
x^2 - 12x - 372 = 48y
x^2 - 12x +36 = 48y + 372 +36
(x-6)^2 = 48y + 408
(x-6)^2 = 48(y + 8.5)
y + 8.8 = (1/48) (x-6)^2
WOW!
Thank you so much! you made more sense than my teacher!
:D
Of course! I'd be happy to help you solve the given parabolas and convert them into the standard form you mentioned, which is the form "x - h = a(y - k)^2".
Let's take one problem at a time.
Problem 1: 3y^2 + 6y + 108x - 969 = 0
To convert this equation into the standard form, follow these steps:
Step 1: Group the variables separately.
(3y^2 + 6y) + (108x - 969) = 0
Step 2: Factor out any common factor from each group, if possible.
3y(y + 2) + 108(x - 9) = 0
Step 3: Divide the coefficient of y by 2 and square the result.
In this case, (2/2)^2 = 1.
Step 4: Add and subtract the value obtained in Step 3 inside the parentheses for the y group only.
3(y^2 + 2y + 1) + 108(x - 9) = 0
Step 5: Simplify the equation inside the parentheses.
3(y + 1)^2 + 108(x - 9) = 0
Step 6: Rearrange the equation according to the standard form.
3(y + 1)^2 = -108(x - 9)
Divide both sides of the equation by -108 to make the coefficient of x equal to 1.
(y + 1)^2 = (-108/3)(x - 9)
Simplify further by dividing -108 by 3.
(y + 1)^2 = -36(x - 9)
Now, the equation is in the desired form, x - h = a(y - k)^2. Therefore, for this problem, h = 9, k = -1, and a = -36.
Let's move on to the second problem:
Problem 2: x^2 + 14x = 44y + 313 = 0
It appears that there is an extra "=" sign in this equation. I will assume that it's a typo, and the equation should be x^2 + 14x = 44y + 313.
Step 1: Group the variables separately.
(x^2 + 14x) = 44y + 313
Step 2: Complete the square for the x group.
To do this, take half of the coefficient of x, which is 14, square it, and add and subtract the result inside the parentheses.
(x^2 + 14x + (14/2)^2 - (14/2)^2) = 44y + 313
Simplify:
(x^2 + 14x + 7^2) - 7^2 = 44y + 313
Step 3: Simplify further the x group.
(x^2 + 14x + 49) - 49 = 44y + 313
Step 4: Rearrange the equation to match the standard form.
(x^2 + 14x + 49) = 44y + 313 + 49
(x^2 + 14x + 49) = 44y + 362
Step 5: Move all the terms to one side of the equation.
(x^2 + 14x + 49) - 44y - 362 = 0
Step 6: To rearrange the equation into the desired form, let's group the terms with the same variable.
(x^2 + 14x + 49) - 44y - 362 = 0
x^2 + 14x + 49 - 44y - 362 = 0
Step 7: Rearrange the terms to match the standard form.
x^2 + 14x - 44y + 49 - 362 = 0
x^2 + 14x - 44y - 313 = 0
Now, the equation is in the form x - h = a(y - k)^2. For this problem, h = -7, k = 0, and a = -44.
I hope this explanation helps you understand the steps involved in converting the given parabolas into the desired form. If you have any further questions, feel free to ask!