Suppose you have two numbers. The difference of the two numbers is 12. The product of the two numbers is 17. Multiply the larger of the two numbers by 100, add 50 times the smaller number to that, round the total to the nearest whole number.
That starts out as an algebra problem with two equations in two unknowns.
x -y = 12
xy = 17
Since 17 is a prime number, x and y will not be integers. Solve by substitution.
x - 17/x = 12
x^2 -12 x -17 = 0
Once you have found out the numbers, complete the final operations. We will be happy to critique your work. You learn by trying.
To solve the quadratic equation x^2 - 12x - 17 = 0, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -12, and c = -17. Substituting these values into the formula, we have:
x = (-(-12) ± √((-12)^2 - 4(1)(-17))) / (2(1))
x = (12 ± √(144 + 68)) / 2
x = (12 ± √212) / 2
Simplifying the expression within the square root, we have:
x = (12 ± √(4 * 53)) / 2
x = (12 ± 2√53) / 2
x = 6 ± √53
So the two possible values for x are 6 + √53 and 6 - √53.
To find the corresponding values of y, we can use the equation xy = 17. Let's take each value of x and solve for y:
For x = 6 + √53:
(6 + √53) * y = 17
y = 17 / (6 + √53)
For x = 6 - √53:
(6 - √53) * y = 17
y = 17 / (6 - √53)
Now, to find the larger number multiplied by 100, we compare the two values of x and choose the larger one. Let's say x = 6 + √53 is larger.
larger number * 100 = (6 + √53) * 100
Next, we add 50 times the smaller number to the result obtained from the previous step:
result = (6 + √53) * 100 + 50 * y
Lastly, round the total to the nearest whole number. Assuming we have the larger number, we round result to the nearest whole number.
To summarize:
1. Solve the quadratic equation x^2 - 12x - 17 = 0 using the quadratic formula.
2. Substitute the values of x into xy = 17 to find the corresponding y values.
3. Compare the two values of x and choose the larger one.
4. Multiply the larger number by 100.
5. Add 50 times the smaller number to the result.
6. Round the total to the nearest whole number.
Please note that the values of x and y may not be integers since 17 is a prime number. Also, ensure you check your calculations and round accurately to get the final answer.