The sum of two numbers is 15 and the sum of their squares is 377. What is the larger number?

4 Answers
Mar 30, 2017

The larger number is #19#

Explanation:

Write two equations with two variables:
#x + y = 15 " and " x^2 + y^2 = 377#

Use substitution to solve:

  1. Solve for one variable # x = 15 - y#

  2. Substitute # x = 15 - y# into the second equation:
    #(15 - y)^2 + y^2 = 377#

  3. Distribute:# (15-y)(15-y) + y^2 = 377#
    #15^2 - 30 y +y^2 + y^2 = 377#
    #255 - 30 y + 2y^2 = 377#

  4. Put in general form #Ax^2 + Bx +C = 0#:
    #2y^2 - 30y +225 - 377 = 0#
    #2y^2 - 30y - 152 = 0#

  5. Factor
    #2(y^2 - 15y - 76) = 0#
    #2(y +4)(y - 19) = 0#
    #y = -4, y = 19#

  6. Check:
    #-4 + 19 = 15#
    #(-4)^2 + 19^2 = 377#

Mar 30, 2017

The larger number is 19.

Explanation:

Since you have two numbers, you must have two equations that relate these numbers to one another. Each sentence provides one equation, if we can translate them properly:

"The sum of two numbers is 15" : #x+y=15#

"The sum of their squares is 377" : #x^2+y^2=377#

Now, we must use the simpler equation to replace one of the unknowns in the more complex equation:

#x+y= 15# means #x=15-y#

Now, the second equation becomes

#x^2 + (15-x)^2 = 377#

Expand the binomial:

#x^2 + 225-30x+x^2=377#

Write in standard from:

#2x^2-30x-152=0#

This can be factored (because the determinant #sqrt(b^2-4ac)# is a whole number.

Might be simpler to just use the quadratic formula, though:

#x=(-b+-sqrt(b^2-4ac))/(2a) = (30+-sqrt((-30)^2-4(2)(-152)))/(2(2))#

#x=(30+-46)/4#

#x=-4# and #x=19# are the answers.

If you check the two answers in the original equations, you will find that both yield that same result! The two numbers we seek are 19 and -4.

That is, if you put #x=-4# into the first equation (#x+y=15#), you get #y=19#.

If you put #x=19# into that equation, you get #y=-4#.

This happens because it does not matter which value we use in the substitution. Both yield the same result.

Mar 30, 2017

#19#

Explanation:

let say the two numbers are #x# and #y#.

#x + y = 15 -> x = 15 -y#

#x^2 + y^2 = 377#

#(x + y)^2 - 2xy = 377#

#15^2 - 2 (15 -y)y = 377#

#225 - 30y + 2y^2 =377#

#2y^2 -30 y - 152 =0#

#(2y + 8)(y - 19) = 0#

#y = -4 and 19#
#x = 19 and -4#
therefore the largest number is #19#

Mar 30, 2017

#19# is the larger number.

Explanation:

It is possible to define both numbers by using only one variable.

The sum of two numbers is #15#.

If one number is #x#, the other is #15-x#

The sum of their squares is #377#

#x^2 + color(red)((15-x)^2) = 377#

#x^2 + color(red)(225 -30x+x^2) -377 =0#

#2x^2 -30x -152 = 0" "larr div 2# to simplify

#x^2 -15x -76 =0#

Find factors of #76# which differ by 15#

#76# does not have many factors, should be easy to find.
#76= 1xx76" "2 xx 38" "color(blue)(4xx19)#

#(x-19)(x+4)=0#

#x = 19 or x = -4#

The two numbers are:

#-4 and 19 #

#16+361 =377#