SOLUTION: What real value of t produces the smallest value of the quadratic t^2 -9t

SOLUTION: What real value of t produces the smallest value of the quadratic t^2 -9t - 36 + 8t^2 + 55t + 41?

Question 1209301: What real value of t produces the smallest value of the quadratic t^2 -9t - 36 + 8t^2 + 55t + 41?
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Answer: -23/9
This is approximately equal to -2.55556 where the 5's go on forever but we have to round at some point.

Explanation

I'll use x in place of t.

x^2 -9x - 36 + 8x^2 + 55x + 41
= (x^2 + 8x^2) + (-9x + 55x) + ( -36 + 41)
= 9x^2 + 46x + 5

The vertex of y = ax^2+bx+c is located at (h,k) where h = -b/(2a)
y = 9x^2 + 46x + 5 has the coefficients a = 9, b = 46, c = 5

h = -b/(2a)
h = -46/(2*9)
h = -46/18
h = (-2*23)/(2*9)
h = -23/9
This is the x coordinate of the vertex.
This x value will make 9x^2 + 46x + 5 as small as possible.
Note that a = 9 is positive, so the parabola opens upward, which places the vertex at the lowest point.

--------------------------------------------------------------------------

Another approach would be to use the quadratic formula to solve 9x^2 + 46x + 5 = 0 to get the roots x = -5 and x = -1/9
I'll leave the scratch work for the student to do.
Or you could factor 9x^2 + 46x + 5 into (x+5)(9x+1) to be able to see the roots easily.

x = -5 and x = -1/9 are where the parabola crosses the x axis.
The midpoint of the x intercepts is the x coordinate of the vertex.
This is due to the parabola's mirror symmetry about the center line.

p,q are the roots
h = x coordinate of the vertex (h,k)
h = (1/2)*(p+q)
h = (1/2)*( -5 + (-1/9) )
h = (1/2)*( -45/9 + (-1/9) )
h = (1/2)*(-46/9)
h = (-46)/(2*9)
h = (-2*23)/(2*9)
h = -23/9

RELATED QUESTIONS
What real value of $t$ produces the smallest value of the quadratic $t^2 -9t -... (answered by ikleyn)

If t is a real number, what is the maximum possible value of the expression -t^2 + 8t -4? (answered by ikleyn)

If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t... (answered by rothauserc)

If t is a real number, what is the maximum possible value of the expression -t^2 + 8t -4... (answered by josgarithmetic,math_tutor2020,mccravyedwin,Edwin McCravy,AnlytcPhil,timofer)

5r=9t+7 what is the value of... (answered by rfer)

33-8t=15 what is the value of... (answered by rfer)

what is the value of t in... (answered by Earlsdon,stanbon,Edwin McCravy)

Find all real values of t that satisfy the equation (t^2 - 13)^2 = 144 +... (answered by Edwin McCravy)

Solve using the quadratic formula:... (answered by josgarithmetic)