SOLUTION: For what constant k is 1 the minimum value of the quadratic 3x^2 - 15x + k over all real values of x? (x cannot be nonreal.)

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: For what constant k is 1 the minimum value of the quadratic 3x^2 - 15x + k over all real values of x? (x cannot be nonreal.)       Log On


   

Question 1088756: For what constant k is 1 the minimum value of the quadratic 3x^2 - 15x + k over all real values of x? (x cannot be nonreal.)
Found 3 solutions by Boreal, Fombitz, ikleyn:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
The vertex is at the minimum for a quadratic where the coefficient of the square term is positive.
-b/2a=15/6=2.5
when x=2.5, y must be 1 in this problem.
substitute
1=3(2.5^2)-15(2.5)+k
1=18.75-37.5+k
19.75=k ANSWER
graph%28300%2C300%2C-10%2C10%2C-10%2C10%2C3x%5E2-15x%2B19.75%29

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Complete the square,
f%28x%29=3x%5E2+-+15x+%2B+k
f%28x%29=3%28x%5E2-5x%29%2Bk
f%28x%29=3%28x%5E2-5x%2B%285%2F2%29%5E2%29%2Bk-3%285%2F2%29%5E2
f%28x%29=3%28x-5%2F2%29%5E2%2B%28k-75%2F4%29
So now the quadratic is in vertex form, the minimum value occurs there,
k-75%2F4=1
highlight%28k=79%2F4%29
.
.
.
.

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
On finding the minimum/maximum of a quadratic function see the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola

in this site, and associated lessons

    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area
    - A farmer planning to fence a rectangular area along the river to enclose the maximal area
    - A rancher planning to fence two adjacent rectangular corrals to enclose the maximal area
    - Using quadratic functions to solve problems on maximizing revenue/profit
    - OVERVIEW of lessons on finding the maximum/minimum of a quadratic function


Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".