3y plus x2 equals 5x-4

Answer 1

What is it that you want done with that equation? The equation alone is not a question.

3y + x2 = 5x - 4

Do you want to know what kind of curve that equation describes? It's a parabola with a maximum y value and no minimum.

Do you want to solve it for y?

y = -(x2 - 5x + 4) / 3

Do you want to solve for x?

x2 - 5x = -3y - 4

x2 - 5x + (5/2)2 = -3y - 4 + (5/2)2

(x - 5/2)2 = -12y/4 - 16/4 + 25/4

(x - 5/2)2 = (9 - 12y)/4

x - 5/2 = ± √(9 - 12y) / 2

x = [5 ± √(9 - 12y)] / 2

Do you want the rate of change of y with respect to x?

dy/dx = (-2x + 5)/3

Do you want to know the vertex of the parabola? Solve its derivative for y:

0 = (-2x + 5) / 3

-2x + 5 = 0

x = 5/2

Then plug that x back into the original equation to find the y coordinate:

y = -(x2 - 5x + 4) / 3

y = -([5/2]2 - 5[5/2] + 4) / 3

y = -(25/4 - 25/2 + 4)/3

y = -25/12 + 50/12 - 16/12

y = 3/4

So the parabola's vertex is at the point (5/2, 3/4)

Do you want the rate of change of x with respect to y?

x = [5 ± √(9 - 12y)] / 2

x = 5/2 ± 1/2(9 - 12y)1/2

dx/dy = ± 1/4(9 - 12y)-1/2 (-12)

dx/dy = ± 3 / √(9 - 12y)

Do you want the indefinite integral of y with respect to x?

∫ -(x2 - 5x + 4) / 3 dx

= -1/3 ∫(x2 - 5x + 4) dx

= -1/3(x3/3 - 5x2/2 + 4x) + C

= -1/3(2x3/6 - 15x2/6 + 24x/6) + C

= -x/18(2x2 - 15x + 25) + C

= -x/18(2x2 - 10x - 5x + 25) + C

= -x(2x - 5)(x - 5)/18 + C

If you were looking for some other answer regarding the given equation, then I would recommend expressing your question a little more clearly.