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• Mar 29th 2013, 12:20 AM
pauliana

Q: The quadratic function which takes the value 41 at x=-2 and the value 20 at x=5 and is minimized at x=2 is

y = __x2 - __x + __

The minimum value of this function is __

I really need to solve this (Worried) . thank you
• Mar 29th 2013, 01:42 AM
emakarov
Let y = ax^2 + bx + c for some unknown a, b and c. Write the given conditions as equations in a, b and c. You'll get three equations in three variables. Recall that the point of minimum of a parabola is -b / (2a).
• Mar 29th 2013, 02:09 AM
pauliana
that will be the equation but i don't get how to compute it. I'm sorry I'm totally clueless :(
• Mar 29th 2013, 02:15 AM
emakarov
Quote:

Originally Posted by pauliana
that will be the equation

What will be the equation?

Quote:

Originally Posted by pauliana
but i don't get how to compute it.

You don't compute an equation; you solve it.

Start by substituting x = -2 into ax^2 + bx + c = 41. Similarly, substitute x = 5 into ax^2 + bx + c = 20. Finally, write the equation saying that the point of minimum, i.e., x = -b / (2a), equals 2.
• Mar 29th 2013, 02:28 AM
pauliana
a(-2)^2 + b(-2) + c = 41 = -4a - 2b + c = 41

a(5)^2 + b(5) + c = 20 = 25a + 5b + c = 20

is this correct?
• Mar 29th 2013, 02:31 AM
emakarov
Correct except that (-2)^2 = 4 and not -4. Negative times negative gives positive. Now you need the third equation, and then you can solve the resulting system of equations.
• Mar 29th 2013, 02:42 AM
pauliana
it this right?

4a - 2b + c = 41
25a + 5b + c = 20
____________________
29a - 3b + 2c = 61
• Mar 29th 2013, 02:45 AM
emakarov
Quote:

Originally Posted by pauliana
it this right?

4a - 2b + c = 41
25a + 5b + c = 20
____________________
29a - 3b + 2c = 61

No, 5b - 2b = 3b, not -3b. Also, you don't gain anything by adding the equations because the resulting equation still has all three variables. It would be better to subtract the equations; then c would be eliminated.

Quote:

Originally Posted by emakarov
Now you need the third equation, and then you can solve the resulting system of equations.

• Mar 29th 2013, 02:54 AM
pauliana
4a - 2b + c = 41
- 25a + 5b + c = 20
_____________________
-21a + 3b =21

right?
wait how about The minimum value of this function is ___ ?
• Mar 29th 2013, 03:08 AM
emakarov
Quote:

Originally Posted by pauliana
4a - 2b + c = 41
- 25a + 5b + c = 20
_____________________
-21a + 3b =21

Hmm, no. The minus in the second equation looks like it acts on 25a only. In any case, -2b - 5b = -7b, so the result should be -21a -7b = 21. You can divide both sides by 7.

Quote:

Originally Posted by pauliana
wait how about The minimum value of this function is ___ ?

Quote:

Originally Posted by emakarov
Recall that the point of minimum of a parabola is -b / (2a).

Quote:

Originally Posted by emakarov
Finally, write the equation saying that the point of minimum, i.e., x = -b / (2a), equals 2.

• Mar 29th 2013, 03:26 AM
pauliana
7 [-21a -7b = 21] 7 will be -3a = 7

then

x = -b
______ = 2
2a

I'm so sorry I'm lost again :(
• Mar 29th 2013, 03:34 AM
pauliana
I found the answer sheet for this question and it says

y = 3x^2 - 12x + 5

then the value of D = -7

I don't get it
• Mar 29th 2013, 03:35 AM
emakarov
First, it is better to avoid notation spanning several lines because this forum does not preserve alignment. Instead of $x=\frac{-b}{2a}=2$, you can simply write x = -b / (2a) = 2.

I don't understand the notation "7 [-21a -7b = 21] 7". Dividing -21a - 7b = 21 by 7 gives -3a - b = 3. The last equation, -b / (2a) = 2, can be multiplied by 2a to get -b = 4a, or 4a + b = 0. Thus, you have

\begin{align*}-3a - b &= 3\\4a + b &= 0\end{align*}

Adding these equations gives a = 3. Substituting this value of a into either of the two equations above gives you b, and knowing a and b you can find c from any of the equations in post #5 (with the correction in post #6).
• Mar 29th 2013, 03:39 AM
emakarov
Quote:

Originally Posted by pauliana
I found the answer sheet for this question and it says

y = 3x^2 - 12x + 5

then the value of D = -7

I don't get it

The minimum value D of this quadratic function is the value at the point of minimum (see the end of post #10).
• Mar 29th 2013, 03:51 AM
pauliana