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
