the sum of two numbers is 20 and the sum of their squares is 272. find the numbers

Question

the sum of two numbers is 20 and the sum of their squares is 272. find the numbers I need help with to solve the equation and getting a statement like what to represent

Answer 1

Hi Jasmine,

 

So we have to translate the words into Math symbols.

 

Sum of two numbers is 20.  This means adding 2 numbers, which we don't know, gets us 20.  So, when we don't know numbers, we use variables and come up with:

 

X + Y = 20

 

Next they say, the sum of their squares is 272.  Same numbers, still the sum, but now it is of their squares, we now we write:

 

X² + Y² = 272

 

Now we have two equations, we can use the first one so solve for one of the variables, I'll choose to solve for X since it is the most common.  To do that, I subtract Y from both sides of the equal sign and end up with:

 

X = 20 - Y

 

Now I take this equation and plug it into the equation with the Squares:

 

(20 - Y)² + Y² = 272

 

We have to do the multiplication here so I'm going to expand it so it is easier to see:

 

(20 - Y)(20 - Y) + Y² = 272

 

So now, we FOIL the first part and end up with:

 

400 - 40Y + Y^{2 +} Y² = 272

 

It is looking like its going to turn into a quadratic, so I will combine the two Y² and bring the 272 over to the left side:

 

2Y² - 40Y + 128 = 0

 

Everything has a factor of 2, so I am going to divide every term by 2, including the 0

 

Y² - 20Y + 64 = 0

 

Now I factor:

 

(Y - 16)(Y - 4) = 0

 

Which gives us the two possible solutions for Y = 16,4

 

We then take those values and plug them back into the first equation and get our possible X values:

 

X = 4,16

 

So we have two possible Solutions:

 

(16,4) or (4,16)

Answer 2

Let x and y represent your two numbers.

 

You know that 

x + y = 20

and

x² + y² = 272

 

You can rewrite the first equation so that

x = 20 - y

and substitute that into the second

 

(20 - y)² + y² = 272

 

expand and you get 

 

400 - 40y + y² + y² = 272

 

2y² - 40y + 128 = 0

2 (y² - 20y + 64) = 0

 

Use the quadratic formula and you get

2(x - 4)(x - 16) = 0

 

Your roots are 4 and 16, so one number is 4 and the other is 16.

 

You should double check your answer

 

4 + 16 = 20

4² + 16² =

16 + 256 = 272

 

Bingo bango bongo, you are good to go!