SOLUTION: 3. Write the equation in standard form, identify, and sketch. 3y^2 + 2x – 5y – 4 = 0 (Hint: complete the square in y)

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: 3. Write the equation in standard form, identify, and sketch. 3y^2 + 2x – 5y – 4 = 0 (Hint: complete the square in y)      Log On


   

Question 129373: 3. Write the equation in standard form, identify, and sketch.

3y^2 + 2x – 5y – 4 = 0
(Hint: complete the square in y)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
3y%5E2+%2B+2x+-+5y+-+4+=+0 Start with the given equation

3y%5E2-5y-4+=+-2x Subtract 2x from both sides

3y%5E2-5y++=-2x%2B+4 Add 4 to both sides

3%28y%5E2-%285%2F3%29y%29++=+-+2x%2B+4 Factor out the leading coefficient 3


Take half of -5%2F3 to get -5%2F6. Now square -5%2F6 to get 25%2F36

3%28y%5E2-%285%2F3%29y%2B25%2F36-25%2F36%29++=+-+2x%2B+4 Complete the square by adding and subtracting 25%2F36. Note: 25%2F36-25%2F36=0

3%28%28y-5%2F6%29%5E2-25%2F36%29++=+-+2x%2B+4 Factor y%5E2-%285%2F3%29y%2B25%2F36 to get %28y-5%2F6%29%5E2

3%28y-5%2F6%29%5E2-25%2F12++=+-+2x%2B+4 Distribute and multiply

3%28y-5%2F6%29%5E2-25%2F12-4++=+-+2x Subtract 4 from both sides

3%28y-5%2F6%29%5E2-73%2F12=+-+2x Combine like terms

%283%28y-5%2F6%29%5E2-73%2F12%29%2F%28-2%29=x Divide both sides by -2 to isolate x

-%283%2F2%29%28y-5%2F6%29%5E2%2B73%2F24=x Divide and reduce

x=-%283%2F2%29%28y-5%2F6%29%5E2%2B73%2F24 Rearrange the equation


So after completing the square, the equation goes from 3y%5E2+%2B+2x+-+5y+-+4+=+0 to x=-%283%2F2%29%28y-5%2F6%29%5E2%2B73%2F24


So the conic section is a parabola that looks like this


Graph of x=-%283%2F2%29%28y-5%2F6%29%5E2%2B73%2F24