Question #9133b

3 Answers
Nov 30, 2017

The two numbers are #10 and 14#.

Explanation:

Given that: the sum of two numbers is 24

If one number is assumed to be #x#, then the other number will be #24-x#

Also given that the product of the two numbers is 140 :

#=> x xx (24- x) =140#

#=> 24x - x^2 = 140#

# => -x^2 +24x -140 = 0 #

#=> x^2 - 24x +140 =0 #-------is a quadratic equation.

To solve it, we need two such numbers, which sum up to give the coefficient of the middle term, i.e.# -24# and their product must be equal to the product of coefficients of first term and last term, i.e. #1 xx 140 =140#.

Two such numbers are# -14 and -10#

#=> x^2 -14x -10x +140=0#

#=> x(x-14) -10(x-14) =0 #

#=> (x-14)(x-10)=0#

#=> x=14 or x=10#

If we take #x=14# the other number is #24-x=24-10#
and
If we take #x= 10# then the other number is #24-10=14#

That means, the two numbers are #10 and 14#.

Nov 30, 2017

Formulate the problem

Explanation:

These kinds of problems are easy to solve when you understand how to formulate them.

Let the numbers be #x# and #y#. Then you have

#x+y = 24#
#xy = 140#

You can solve these in a couple of ways.

Method 1.

#(x-y)^2 = (x+y)^2-4xy#

Then you have

#(x-y)^2 = 24^2-4\times 140#
#(x-y)^2 = 576 - 560#
#(x-y)^2 = 16#
#(x-y) = \pm 4#

Now you have 2 scenarios. #x-y = 4#, #x-y = -4#. Based on which of the equation is suitable for you, you can solve either of these and get a solution.

Solve #x+y =24# and #x-y = 4# then you get #x=14, y= 10#
Solve #x+y = 24# and #x-y = -4# then you get #x=10, y = 14#

Method 2

Substitute and calculate.

#x+y = 24# then #y = 24-x#. Substitute in the other equation

#xy = 140#
#x(24-x) = 140#
#-x^2+24x=140#
#x^2-24x+140 = 0#
#x^2-14x-10x+140 = 0#
#x(x-14)-10(x-14) = 0#
#(x-14)(x-10) = 0#
#x = 14,10#

Then you will get #y = 10,14# after substituting back in #y = 24-x#.

Nov 30, 2017

10 and 14

Explanation:

There must be two equation to solve for two variables.

Let x equal one number
Let y equal the other number

One equation is

# x + y = 24 #

The other equation is

# x xx y = 140#

Solve the first equation for y

# x +y = 24# subtract x from both sides

#x -x + y = 24 - x # this gives

# y = 24 -x#

Substitute this value into the second equation.

#( 24-x) xx x = 140# distribute the x across the parenthesis

# 24 x - x^2 = 140# Subtract 24 x and add x^2

# 24x - 24x - x^2 + x^2 = 140 - 24x + x^2# This results in

# x^2 - 24x + 140 = 0 # Factor this trinomial into binomials

# ( x - 10) xx ( x -14) = 0 #

Solve for x in both binomials

# x -10 = 0 # add 10 to both sides

# x - 10 + 10 = 0 + 10# which gives

# x = 10#

# x -14 = 0 # add both 14 to both sides

# x - 14 + 14 = 0 + 14 # which gives

# x = 14#

The two numbers are 10 and 14