I am having a math problem( quadratic functions)Please can someone take a look??

Question

here is the problem?
One number is 10 larger than another. What is the smallest possible value for the sum of their squares? What are the two numbers?

please explain the steps to this math problem, so that I understand so I guess in the simplest terms possible. Thank you in advance.

Answer 1

Hi,

Suppose one number is "x". Then if the other number is 10 larger than the first number, it is "x + 10".

The square of those 2 numbers would be x^2 and (x + 10)^2, so their sum would be x^2 + (x + 10)^2. If we set that equal to y, we get:

y = x^2 and (x + 10)^2

Multiplying that out and combining like terms, we get:

y = x^2 + ( x + 10)(x + 10)
y = x^2 + x^2 + 10x + 10x + 100
y = 2x^2 +20x + 100

This is a quadratic equation whose graph is a parabola, opening up because the x^2 term is positive. It has a minimum value for its sum at its vertex. the vertex is on the axis of symmetry, which is found by the formula x = -b/(2a). For our equation b = 20 and a = 2, so x = -b/(2a) is
x = -20/(2*2) or x = -20/4 or x = -5.

If the first number is -5 then the other number is x + 10 or 5.

So your numbers are -5 and 5.

I hope that helps you!!! :-)

Answer 2

I'm not sure if you're using calculus or not so I will explain it that way first and then I will explain it intuitively.

Look at the question like a function.
Two numbers one is 10 greater than the other. So w have:
x and x+10
We're looking at the sum of their squares:
f(x) = x^2 + (x+10)^2
= 2*x^2 + 20*x + 100
now take the derivative:
f'(x) = 4*x +20
We want to find the minimum so set the derivative equal to 0 and solve for x
0 = 4*x + 20
-20 = 4*x
-5 = x and then 5 = x+10
the sum of their squares is 50.

Intuitively you should see that the square of a number increase rapidly the farthe it gets from 0 so what you want to do is keep both numbers as close to 0 as possible. Just split the difference and make one 5 and the othe -5 you can see if you deviate from this in either direction the sum of the squares increases.

Answer 3

Follow the bouncing ball.

One number is larger than another. Let another=x, then one number is x+10
What is the smallest possible value for the sum of their squares. For "another", the SOS = x^2.
For "one number", the SOS = x^2 + 20x + 100.
So add them together and we have 2x^2+20x+100. Since it didn't state that the sum of the squares is a perfect square, we can choose the minimum positive integers 1 and 11.
Then the SOS= 122.

Answer 4

x = 10 + y

we want to find the smallest possible number that will fit
x^2 + y^2
and also the first equation... so let's solve this like a system of equations... substitute 10 + y in for x...
(10 + y)^2 + y^2 =
100 + 20y + y^2 + y^2 = 100 + 20y + 2y^2

and we want the minimum, so let's take the derivative...

y' = 20 + 4y
and set it equal to 0

20 + 4y = 0
4y = -20
y = -5
and if y = -5, then x = 5

so our 2 numbers are 5 and -5

I'm not sure if there's a way to do this without calculus... I suppose you could do trial and error.

Answer 5

set up your 2 circumstances (bear in strategies that the area wherein the automobile travels keeps to be consistent even though its speed and time adjustments). ---------------- Case a million) prevalent speed speed = x Distance = a hundred and five Time = y Case 2) bigger speed speed = x + 10 Distance = a hundred and five Time = y - a million/4 OR (4y - a million)/4 Now type your 2 simultaneous equations (making use of speed = distance/time): x = a hundred and five/y x + 10 = a hundred and five / (4y - a million)/4 Simplify the complicated fraction: x + 10 = 420 / (4y - a million) ------------------- Now replace a hundred and five/y for x in the 2nd equation: a hundred and five/y + 10 = 420 / (4y - a million) Multiply each little thing via y(4y - a million) to cancel off all denominators: a hundred and five(4y - a million) + 10y(4y - a million) = 420y Divide each little thing via 5 to simplify the equation: 21(4y - a million) + 2y(4y - a million) = 84y develop all the brackets: 84y - 21 + 8y^2 - 2y = 84y Subtract 84y from the two sides and type from optimal potential to lowest: 8y^2 - 2y - 21 = 0 component it out: (2y + 3)(4y - 7) = 0 this provides the potential of the time the automobile travelled in being the two -3/2 hours or 7/4 hours. Now for sure you could not return and forth back in time so the time wherein the automobile travelled can in ordinary terms be 7/4 hours. Now which you realize what the time is discover that to discover the value (making use of speed = distance/time): x = a hundred and five / (7/4) x = a hundred and five * 4/7 x = 60 the unique speed of the automobile became into 60mph.

Answer 6

one number is x
the other number is (x + 10)

Theior squares are x^2

and

(x+10)^2 whcih equals x^2 + 20x + 100 (use FOIL).

The sum of the squares is:

x^2 + (x^2 + 20x + 100) = 2x^2 + 20x +100

The vertex (minimum) of this parabola occurs at x = -b/2a which is -20/4 = -5

Substitute -5 into the equation to get the minimum:

25.

the two numbers are x and (x +10), since x = -5 the two numbers are -5 and 5.

Answer 7

what is quadratic functions