Question

Question #9133b

Answer 1

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`.

Answer 2

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`.

Answer 3

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