Find the minimum value of the quadratic function. Y=(3x-4)(9x+2)?

Question

I tried to foil this out and then CTS to put it in Vertext form but it got some funky numbers. What's the eaisest way to do this problem and what do you get.

Thanx,

Alice

Answer 1

Several ways to solve:
1) Foil, and take the derivative.
y = 27x^2 - 30x - 8

set the derivative to 0: 54x - 30 = 0
x = 30/54 = 5/9

or

2)
If you don't know calculus, here is an easier way:
A parabola is symetrical
The minimum will be the half way point between the two x-intercepts
The two x-intercepts are:
(3x-4) = 0 x = 4/3
(9x+2) = 0 x = -2/9

The average of those 2 numbers =
(4/3 - 2/9)/2 = 5/9


Once you have your x, plug into the equation
y = (3*(5/9) - 4)(9*5/9 + 2)
y = (-7/3)(7) = -49/3

(I did this in my head, so I would double check the work)

Answer 2

Foil it and take the derivative. Find out where y' = 0 (that will be the vertex).

y = 27x^2 - 30x - 8
y' = 54x - 30
0 = 54x - 30
30 = 54x
5/9 = x

At x = 5/9, y = 27(5/9)^2 - 30(5/9) - 8 = 27(25/81) - 150/9 - 8 = 25/3 - 50/3 - 24/3 = -49/3

The vertex is at (5/9, -49/3) and the minimum value is -49/3.

Answer 3

How does this quadratic function open? It opens up because the a coefficient is effective. the position does it bypass the x-axis? It crosses the x axis the position y = 0 or x(x-4) = 0 or x= 0 and x=4 so the vertex lies at x=2 halfway between x=2 y=-4

Answer 4

there is calculus way using d(fg)/dx = fdg/dx+gdf/dx
dy/dx = (3x-4)9+3(9x+2) = 54x -30 so x = 30/54 or 5/9
put the value of x and calculate y
not it could be minimum or maximum
2nd derivative = 54 positive so it is minimum

algebra way
y = 27x^2-30x-8
= 3(9x^2-10x) -8(convert the expression in parenthesis to whole square in next line
= 3((3x)^2)-2(3x)(5/3)+(5/6)^2)-[3(5/6)^2+8]
= 4(3x-5/3)^2 -(75/36+8)
it is minimum when 3x = 5/3 or x= 5/9 and value of y = 8+25/12 or 10 and 1/12

Answer 5

you can use calculus method as shown by some of the people above.

or try the following method

y = 27x^2 - 30x - 8

a=27
b= -30
c= -8

since a is positive, minimum is at the vertex.

vertex is at x = {-b}/ {2a} = 30/54 = 5/9

when x = 5/9

y = -49/3

minimum value of y is { - 49 / 3 }

Answer 6

When you foil you should get 27x^2 + 6x - 36x - 8
which is 27x^2 -30x -8

In quadratic formula:
30+/- the square root of (30^2 - 4(27)(-8)) all over 2(27)

simpiler terms: 30+/- the square root of (900+864) all over 54

even simpiler: 30+/- (42) all over 54

the answer is (-4 1/2, 4/3)

Answer 7

1) multiply it out; y=27x2-30x-8
2) reformat; y=9 ( x - 5/3) 2 - 10 7/9
3) minimum value, since a is positive, then minimum value = y coordinate of tip
4) tip=(5/3, -97/9)
5) answer = -97/9

Answer 8

Y=27x^2 -30x-x
by calculus. x=10/9 is minimum and then Y is -28. those are minimum value

Answer 9

dY/dx=27x-30 and d^2y/dx^2=27>O then 27x-30=O and x is 10/9. d^2y/dx^2>0 so x=10/9 is minimum then Y=-28 is the minimum

Answer 10

-1(?) I guess.I don't know. i'm just sixth grade.