The sum of two positive numbers is 42 and their product is 425. What are the two numbers.
How would I find the answer to this?
a + b = 42 ... a = 42 - b
a * b = 425
substituting ... b (42 - b) = 425 ... 42 b - b^2 = 425 ... b^2 - 42 b + 425 = 0
factor or use the quadratic formula
... the two roots equal a and b
To find the two numbers, let's start by assigning variables to them. Let's call the first number "x" and the second number "y".
We know that the sum of the two numbers is 42, so we can write the equation:
x + y = 42
We also know that the product of the two numbers is 425, so we can write the equation:
xy = 425
Now we have a system of two equations with two variables. We can solve this system using a method called substitution or elimination.
Let's solve it using substitution.
From the first equation, we can isolate one of the variables. Let's choose y. We can rewrite the first equation as:
y = 42 - x
Now substitute this value of y into the second equation:
x(42 - x) = 425
Expand the equation:
42x - x^2 = 425
Rearrange the equation to form a quadratic equation:
x^2 - 42x + 425 = 0
Now we can solve this quadratic equation. We can either factorize it or use the quadratic formula.
Let's factorize it:
(x - 17)(x - 25) = 0
Setting each factor to zero gives two possible values for x:
x - 17 = 0 or x - 25 = 0
Solving each equation for x gives:
x = 17 or x = 25
Now substitute these values of x back into the equation y = 42 - x to find the corresponding values of y:
For x = 17, y = 42 - 17 = 25
For x = 25, y = 42 - 25 = 17
Therefore, the two numbers are 17 and 25.
To find the two numbers, we can set up a system of equations based on the given information and solve for the variables. Let's call the two numbers x and y.
We are given two conditions:
1. The sum of the numbers is 42:
x + y = 42
2. The product of the numbers is 425:
xy = 425
Now, we have a system of two equations with two variables. We can solve this system using various methods like substitution, elimination, or graphing. Let's use the substitution method:
From equation 1, we can solve for x:
x = 42 - y
Substituting this value of x into equation 2, we have:
(42 - y)y = 425
Expanding and rearranging the equation:
42y - y^2 = 425
y^2 - 42y + 425 = 0
Now we have a quadratic equation. We can solve it by factoring, completing the square or using the quadratic formula. In this case, let's use factoring:
(y - 25)(y - 17) = 0
Setting each factor equal to zero:
y - 25 = 0 or y - 17 = 0
Solving for y:
y = 25 or y = 17
Now plugging these values back into equation 1 to solve for x:
When y = 25:
x = 42 - 25
x = 17
When y = 17:
x = 42 - 17
x = 25
Therefore, the two numbers are 17 and 25.