If the sum of two numbers is 24 and the product is 140 what are the two numbers

The two numbers are 10 and 14

Step-by-step explanation:

To find the sum of two numbers whose sum is 24 and product 140;

we let x and y to be the two numbers. From the question, the sum of the two numbers is 24; that is x + y = 24 (1)

Also, from the question, the product of the two number is 140,

that is; xy = 140 (2)

So we are now having a system of equations, we are going to use substitution method to solve this.

From equation (1)

x + y = 24

we want to make x subject of the formula, so we will simply subtract y from both-side of the equation;

x  + y-y  = 24 - y

x = 24 -y (3)

substitute equation (3) in equation  (2)

xy = 140

(24-y)y = 140

open the bracket;

24y - y²  = 140

we can re-arrange the equation to become;

y²  - 24y  + 140 = 0

This is now a quadratic equation, we can solve by completing the square method, we simply find two numbers whose sum will give us -24 and whose product will give us 140 and then replace -24y by the two numbers. The two numbers are -14 and -10.

so, we are going to replace -24y by -14y and  -10y

Thus;

y²  - 14y - 10y  + 140 = 0

We will now factorize

y(y-14) -10(y-14)=0

(y-10)(y-14)=0

Either y-10=0

          y =10

OR

y - 14 =0

y=14

Either y= 10 or y = 14

Substituting  for y into our equation(1)

x + y = 24

when  y = 10

x + 10 = 24

x =24-10

x =14

When y=14

x + 14 = 24

x = 24-14

x = 10

Therefore, the two numbers are 10 and 14

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