What is the vertex form of #3y=-3x^2 - 7x -2?

Question

Eli Assouline · Accepted Answer

#color(green)(y= (x-7/6)^2-73/36)#
Notice I have kept it in fractional form. This is to maintain precision.

Divide through out by 3 giving: #y=x^2-7/3x-2/3# British name for this is: completing the square You transform this into a perfect square with inbuilt correction as follows: #color(brown)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")# #color(brown)("Consider the part that is: "x^2-7/3x)# #color(brown)("Take the"(-7/3)"and halve it. So we have"1/2 xx(-7/3)=(-7/6))# #color(brown)("~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~")# Now write: #y-> (x-7/6)^2-2/3# I have not used the equals sign because an error has been introduced. Once that error is removed we can then start to use the = sign again. #color(white)(xxxxxxxx)"----------------------------------------------"# #color(red)(underline("Finding the introduced error"))# If we expand the brackets we get: #color(brown)(y->x^2- 7/3 xcolor(blue)(+(7/6)^2)-2/3# The blue is the error. #color(white)(xxxxxxxx)"----------------------------------------------"# #color(red)(underline("Correction for the introduced error"))# We correct for this by subtracting the same value so that we have: #color(brown)(y->x^2- 7/3 xcolor(blue)(+(7/6)^2-(7/6)^2)-2/3# Now lets change the bit in green back to where it came from: #color(green)(y->x^2- 7/3 x+(7/6)^2color(blue)(-(7/6)^2-2/3))# Giving: #color(green)(y= (x-7/6)^2)color(blue)(-(7/6)^2-2/3# The equals sign (=) is now back as I have included the correction. #color(white)(xxxxxxxx)"----------------------------------------------"# #color(red)(underline("Finalising the calculation"))# Now we can write: #y= (x-7/6)^2-(49/36)-2/3# #2 1/36# #color(green)(y= (x-7/6)^2-73/36)#

Kennedy Adams · Answer

undefined To find the vertex form of the quadratic equation 3y = -3x^2 - 7x - 2, follow these steps:

1. Divide both sides of the equation by 3 to isolate y: y = (-3/3)x^2 - (7/3)x - (2/3).
2. Rewrite the equation in the form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
3. Complete the square for the x-terms:
   - For the coefficient of x^2, we have a = -1, so we get (-1/3)(x^2 + (7/3)x) - (2/3).
   - To complete the square, take half of the coefficient of x, square it, and add it inside the parentheses: (-1/3)(x^2 + (7/3)x + (7/6)^2 - (7/6)^2) - (2/3).
4. Simplify the expression inside the parentheses: (-1/3)((x + 7/6)^2 - (49/36)) - (2/3).
5. Distribute the -1/3: (-1/3)(x + 7/6)^2 + 49/108 - (2/3).
6. Combine the constants: (-1/3)(x + 7/6)^2 + (49/108) - (72/108).
7. Combine the constants further: (-1/3)(x + 7/6)^2 - (23/108).

So, the vertex form of the quadratic equation 3y = -3x^2 - 7x - 2 is y = (-1/3)(x + 7/6)^2 - 23/108.