What Determines the Minimum Value in a Quadratic Equation?

[Wildcat]

Homework Statement

4. If x1, x2 are two real number roots of real number coefficient quadratic equation:
$$x^2 -2mx + m + 2 =0$$
Solve the following questions:
(1) What are the values of m so that x1=x2?
(2) What are the value(s) so that $$(x1)^2 + (x2)^2 $$
has minimum value? What is the minimum?

Homework Equations

The Attempt at a Solution

I substituted x1 into the quadratic for x and then substituted x2 into the quadratic for x and set them = to each other. I then solved for m and got
m= (x1 + x2)/2

I'm not sure what the 2nd question means. Can anyone help?
Thanks

 

[Office_Shredder]

When x₁ = x₂, m is going to be some specific number, like 3.5, or [itex] \sqrt{2}[/itex] or something. Can you tell us what that number is and show us how you calculated it (or if not, show us how you got what you got).

 

[Chestermiller]

For a quadratic equation of the form ax²+bx+c=0, did they teach you what the sum of the roots is equal to in terms of a, b, and c? Did they teach you what the product of the roots is equal to?

For part 1, can you use the quadratic formula to express the roots of the equation in terms of m?

 

[Wildcat]

substitute x1 into the equation to get
x1^2 -2mx1 +m+2
Substitute x2 to get
x2^2 -2mx2 +m+2
set them = to each other and solve for m
x1^2 -2mx1 + m + 2 = x2^2 -2mx2 +m + 2
so
x1^2 - 2mx1 = x2^2 -2mx2
then
x1^2 -x2^2 = m2(x1 - x2)
then factor and divide to get
m=(x1 + x2)/2

 

[Chestermiller]

substitute x1 into the equation to get
x1^2 -2mx1 +m+2
Substitute x2 to get
x2^2 -2mx2 +m+2
set them = to each other and solve for m
x1^2 -2mx1 + m + 2 = x2^2 -2mx2 +m + 2
so
x1^2 - 2mx1 = x2^2 -2mx2
then
x1^2 -x2^2 = m2(x1 - x2)
then factor and divide to get
m=(x1 + x2)/2

Are you familiar with the formula:
[tex]x=\frac{-b±\sqrt{b^2-4ac}}{2a}[/tex]
In terms of m, what are a, b, and c in your equation. Using the above equation, what value of m makes the two roots equal to one another?

 

[Wildcat]

For a quadratic equation of the form ax²+bx+c=0, did they teach you what the sum of the roots is equal to in terms of a, b, and c? Did they teach you what the product of the roots is equal to?

For part 1, can you use the quadratic formula to express the roots of the equation in terms of m?

OK let me try that. add the roots and multiply by a to get b (keep the sign of b)and multiply the roots and multiply by a to get c?

 

[Wildcat]

Are you familiar with the formula:
[tex]x=\frac{-b±\sqrt{b^2-4ac}}{2a}[/tex]
In terms of m, what are a, b, and c in your equation. Using the above equation, what value of m makes the two roots equal to one another?

so a=1, b=-2m and c=m+2
then
using the formula you get
m±√m^2 - m + 2

??

 

[mfb]

so a=1, b=-2m and c=m+2
then
using the formula you get
m±√m^2 - m + 2

??

There are brackets missing. The general idea is good, and a,b,c are right.
After you fixed the brackets: Which value of m leads to the same value for x1 and x2?

 

[Chestermiller]

so a=1, b=-2m and c=m+2
then
using the formula you get
m±√m^2 - m + 2

??

You made a mistake in algebra. That should be a minus 2 under the square root.

 

[Saitama]

Why you ask the OP to calculate the roots? The given questions doesn't require that.

For part 1, it can be easily solved by investigating the discriminant of the given quadratic.

For part 2, rewrite ##(x_1)^2+(x_2)^2## as ##(x_1+x_2)^2-2x_1x_2##. Its quite easy to solve from here.

I am sorry if I said something wrong.

 

[Chestermiller]

Why you ask the OP to calculate the roots? The given questions doesn't require that.

For part 1, it can be easily solved by investigating the discriminant of the given quadratic.

For part 2, rewrite ##(x_1)^2+(x_2)^2## as ##(x_1+x_2)^2-2x_1x_2##. Its quite easy to solve from here.

I am sorry if I said something wrong.

Thanks Panav-Arora. Both these suggestions are what I originally had in mind. But the OP seemed to be struggling too much, so I thought I would try to make it easier on him. I thought that having him examine the solution using the quadratic formula would lead him to realizing that the discriminant should be set equal to zero.

Part 2 is obviously a question in which he should have been thinking in terms of the sum of the roots, -b/a, and the product of the roots c/a. This must be what is being covered in his course right now. But he seemed to have no idea about this either. The equation you wrote above is exactly what I was thinking of.

Chet

 

[Ray Vickson]

Homework Statement

4. If x1, x2 are two real number roots of real number coefficient quadratic equation:
$$x^2 -2mx + m + 2 =0$$
Solve the following questions:
(1) What are the values of m so that x1=x2?
(2) What are the value(s) so that $$(x1)^2 + (x2)^2 $$
has minimum value? What is the minimum?

Homework Equations

The Attempt at a Solution

I substituted x1 into the quadratic for x and then substituted x2 into the quadratic for x and set them = to each other. I then solved for m and got
m= (x1 + x2)/2

I'm not sure what the 2nd question means. Can anyone help?
Thanks

Since you have already done the question, I can now come in with a simpler solution.
(1) the quadratic ##q(x) = 0## has a repeated root only if it is of the form ##q(x) = (x-r)^2##, or ##q(x) = x^2 - 2rx + r^2##. So in your case you need ##m + 2 = m^2##.
(2) If ##x_1, x_2## are roots of ##x^2 = 2mx - m-2## then ##x_1^2 + x_2^2 = 2m(x_1 + x_2) - 2m - 4##, and ##x_1 + x_2 = 2m##. Thus, ##x_1^2 + x_2^2 = 4m^2 - 2m - 4.## This has the form y(y-1)-4, where y = 2m. Graphically, we see that the minimum of y(y-1) occurs at y = 1/2, giving m = 1/4. The minimum value of ##x_1^2 + x_2^2## is negative, meaning that x_1 and x_2 are complex.

 

[Wildcat]

You made a mistake in algebra. That should be a minus 2 under the square root.

Yes I did make an error. Thank you.

 

[Wildcat]

You made a mistake in algebra. That should be a minus 2 under the square root.

Ok after I fixed my error and set it = to 0 I get m= 2 or -1
Right??

 

[mfb]

Looks good.

 

[Wildcat]

Since you have already done the question, I can now come in with a simpler solution.
(1) the quadratic ##q(x) = 0## has a repeated root only if it is of the form ##q(x) = (x-r)^2##, or ##q(x) = x^2 - 2rx + r^2##. So in your case you need ##m + 2 = m^2##.
(2) If ##x_1, x_2## are roots of ##x^2 = 2mx - m-2## then ##x_1^2 + x_2^2 = 2m(x_1 + x_2) - 2m - 4##, and ##x_1 + x_2 = 2m##. Thus, ##x_1^2 + x_2^2 = 4m^2 - 2m - 4.## This has the form y(y-1)-4, where y = 2m. Graphically, we see that the minimum of y(y-1) occurs at y = 1/2, giving m = 1/4. The minimum value of ##x_1^2 + x_2^2## is negative, meaning that x_1 and x_2 are complex.

I took the derivative of 4m^2 - 2m -4 to get m=1/4. I showed this to my instructor and this is the reply I got.

Good observation!

then, you just consider m>=2 or m<=-1, and find the minimum value of function y=4m^2-2m-4. It is solvable

I'm not sure what he means by this. I looked at graphs with m=3,4,5, -1,-2 and the minimum keeps getting lower. Any advice?

 

[Mark44]

I took the derivative of 4m^2 - 2m -4 to get m=1/4.

Since you posted this problem in the Precalculus section, I am assuming that you are taking a class that comes before calculus. Using a technique from a later class is usually not appropriate, and in this case, not necessary.

The graph of your function is a parabola that opens up, so its lowest point is at its vertex. You should be able to complete the square to write your equation as y = 4(x - h)² + k. In this form, you can see by inspection that the vertex (low point) is at (h, k).

I showed this to my instructor and this is the reply I got.

Good observation!

then, you just consider m>=2 or m<=-1, and find the minimum value of function y=4m^2-2m-4. It is solvable

I'm not sure what he means by this. I looked at graphs with m=3,4,5, -1,-2 and the minimum keeps getting lower. Any advice?

See above.