What 2 numbers when multiplied you get 21.25, but when added you get 11
To find the two numbers, we can set up the following equations:
Let's say the two numbers are x and y.
Equation 1: x * y = 21.25
Equation 2: x + y = 11
From Equation 2, we can express x in terms of y: x = 11 - y.
Substituting this expression for x into Equation 1, we have:
(11 - y) * y = 21.25
Expanding this equation gives us:
11y - y^2 = 21.25
Rearranging the equation:
y^2 - 11y + 21.25 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. However, we will use factoring to find the roots:
(y - 5)(y - 6.25) = 0
Setting each factor equal to zero:
y - 5 = 0 or y - 6.25 = 0
y = 5 or y = 6.25
Now that we have the values of y, we can substitute them back into Equation 2 to find the corresponding values of x:
When y = 5:
x + 5 = 11
x = 11 - 5
x = 6
When y = 6.25:
x + 6.25 = 11
x = 11 - 6.25
x = 4.75
The two numbers are 6 and 5 (or 6.25 and 4.75) since they multiply to 21.25 and add up to 11.
Let's solve this step by step.
Step 1: Assume the two numbers as variables. Let's call them x and y.
Step 2: According to the problem, when these two numbers are multiplied, we get 21.25. So we can write the equation as:
x * y = 21.25
Step 3: Also, when these two numbers are added, we get 11. So we can write another equation as:
x + y = 11
Now we have a system of two equations that we need to solve simultaneously.
Step 4: Let's solve the second equation for one of the variables. Let's solve for y:
y = 11 - x
Step 5: Substitute this value of y in the first equation:
x * (11 - x) = 21.25
Step 6: Simplify the equation:
11x - x^2 = 21.25
Step 7: Rearrange the equation to make it quadratic:
x^2 - 11x + 21.25 = 0
Step 8: Solve this quadratic equation by factoring, completing the square, or using the quadratic formula. The equation can be factored as:
(x - 5)(x - 6.25) = 0
Step 9: Set each factor equal to zero and solve for x:
x - 5 = 0 --> x = 5
x - 6.25 = 0 --> x = 6.25
Step 10: Now substitute these values back into the equation y = 11 - x to find the corresponding values of y:
For x = 5: y = 11 - 5 = 6
For x = 6.25: y = 11 - 6.25 = 4.75
So, the two numbers that satisfy the given conditions are 5 and 6.25.
To find two numbers that satisfy the given conditions, we can set up a system of equations. Let's call the two numbers x and y.
Given that the product of the two numbers is 21.25, we can write the equation:
x * y = 21.25 -- Equation 1
Given that the sum of the two numbers is 11, we can write the equation:
x + y = 11 -- Equation 2
Now we can solve this system of equations to find the values of x and y.
First, we can rearrange Equation 2 to express one variable in terms of the other. Let's solve for y:
y = 11 - x
Now substitute this expression for y in Equation 1:
x * (11 - x) = 21.25
Expand the equation:
11x - x^2 = 21.25
Rearrange the equation to form a quadratic equation:
x^2 - 11x + 21.25 = 0
To solve this quadratic equation, we can try factoring or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, the coefficients are:
a = 1, b = -11, c = 21.25
Substitute these values into the quadratic formula:
x = (-(-11) ± √((-11)^2 - 4*1*21.25)) / (2*1)
= (11 ± √(121 - 85)) / 2
= (11 ± √36) / 2
= (11 ± 6) / 2
Now we have two possible solutions for x:
x1 = (11 + 6) / 2 = 17 / 2 = 8.5
x2 = (11 - 6) / 2 = 5 / 2 = 2.5
So, the two numbers are approximately 8.5 and 2.5.