SOLUTION: How would I write -4x^2 + 9y^2 + 32x + 36y - 64 = 0 in standard form? What are the steps?

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Question 1106320: How would I write -4x^2 + 9y^2 + 32x + 36y - 64 = 0 in standard form? What are the steps?
Found 3 solutions by Boreal, greenestamps, ikleyn:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
-4x^2 + 9y^2 + 32x + 36y - 64 = 0 ;
put positive square first and move constant to right side. Group x and y
9y^2+36y-4x^2+32x=64. Factor
9(y^2+4y)-4(x^2-8x)=64. Complete the square and add constants to right side
9(y^2+4y+4)-4(x^2-8x+16)=64+36-64
9(y+2)^2-4(x-4)^2=36. Divide both sides by 36
(y-(-2))^2/4-(x-4)^2/9=1, write in the form of (y-k)^2/b^2-(x-h)^2/a^2=1, equation of hyperbola.






Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


(1) complete the squares in both x and y; and
(2) divide by the appropriate constant to get "1" on the right hand side.

-4x%5E2+%2B+9y%5E2+%2B+32x+%2B+36y+-+64+=+0 [original equation]
-4x%5E2%2B32x%2B9y%5E2%2B36y+=+64 [group the x terms and y terms; move constant to other side]
-4%28x%5E2-8x%29%2B9%28y%5E2%2B4y%29+=+64 [factor out the leading coefficients for both x and y]
-4%28x%5E2-8x%2B16%29%2B9%28y%5E2%2B4y%2B4%29+=+64-4%2816%29%2B9%284%29+=+36 [complete the squares in x and y, adding the same quantities to both sides of the equation]
%28-4%2F36%29%28x-4%29%5E2%2B%289%2F36%29%28y%2B2%29%5E2+=+1 [divide by the constant on the right to make the right side "1"]
-%28x-4%29%5E2%2F9%2B%28y%2B2%29%5E2%2F4+=+1 [simplify and re-format]
%28y%2B2%29%5E2%2F4-%28x-4%29%5E2%2F9+=+1 [rearrange terms to get positive term first on left hand side]
%28y%2B2%29%5E2%2F2%5E2-%28x-4%29%5E2%2F3%5E2+=+1 [write in final standard form]

The equation is of a hyperbola with center (4,-2); asymptotes with slopes 2/3 and -2/3; branches opening upward and downward.

A graph: 2 branches of the hyperbola (red, green); asymptotes (blue, purple). The center is at the intersection of the asymptotes.

Answer by ikleyn(52802) About Me  (Show Source):
You can put this solution on YOUR website!
.
It is a typical problem on conical sections.

See the lessons in this site, where you will find all instructions and explanations on how to solve such problems.

Learn it once for all.

    - Hyperbola definition, canonical equation, characteristic points and elements

    - Standard equation of a hyperbola
    - Identify elements of hyperbola given by its standard equation
    - Find the standard equation of a hyperbola given by its elements

    - General equation of a hyperbola
    - Transform general equation of a hyperbola to the standard form by completing the square
    - Identify elements of a hyperbola given by its general equation

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topics
"Conic sections: Hyperbolas. Definition, major elements and properties. Solved problems".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.