Hello,
We are asked to solve the following problem using the quadratic equation:
The sum of two numbers is nine. If the square of one number is three less than the other number, what are the numbers?
I know what the answer is supposed to be, which I'll write at the bottom. My question is this: there seems to be a second way to develop the answer - which doesn't work - but I don't understand why and want to ask for someones perspective. Can someone see what my error is? At the top is "STEPS CONTAINING AN ERROR", and at the bottom is, "STEPS TO CORRECT ANSWER".
STEPS CONTAINING AN ERROR
The sum of two numbers is nine. If the square of one number is three less than the other number, what are the numbers?
Let "x" equal the first number.
Let "y" equal the second number.
x + y = 9, and thus: x = 9 - y
x² = y - 3
Substitute (9 - y) for "x" in the above equation:
(9 - y)² = y - 3
9² - y² = y - 3
81 - y² = y - 3
y² + y - 3 - 81
y² + y - 84
Using the above for the quadratic equation, with: a = 1, b = 1, c = -84
y = [-b ± √(b² - 4ac)] / (2a)
y = [-1 ± √(1 - 4(1)(-84))] / [2(1)]
y = (-1 ± 18.36) / (2)
1st iteration of answer:
y = (-1 + 18.36) / (2), y = 8.68
x = 9 - y, x = 0.32
2nd iteration of answer:
y = (-1 - 18.36) / (2), y = -9.68
x = 9 - y, x = 18.68
STEPS TO CORRECT ANSWER
Let "x" equal the first number.
Let "y" equal the second number.
x + y = 9, and thus: y = 9 - x
x² = y - 3
Substitute (9 - x) for "y" in the above equation:
x² = 9 - x - 3
x² + x - 6
Using the above for the quadratic equation, with: a = 1, b = -1, c = -6
x = [-b ± √(b² - 4ac)] / (2a)
x = [-1 ± √(1 - 4(1)(-6))] / [2(1)]
x = (-1 ± 5) / (2)
1st iteration of answer:
x = (-1 + 5) / (2), x = 2
y = 9 - x, y = 7
2nd iteration of answer:
x = (-1 - 5) / (2), x = -3
y = 9 - x, y = 7
We are asked to solve the following problem using the quadratic equation:
The sum of two numbers is nine. If the square of one number is three less than the other number, what are the numbers?
I know what the answer is supposed to be, which I'll write at the bottom. My question is this: there seems to be a second way to develop the answer - which doesn't work - but I don't understand why and want to ask for someones perspective. Can someone see what my error is? At the top is "STEPS CONTAINING AN ERROR", and at the bottom is, "STEPS TO CORRECT ANSWER".
STEPS CONTAINING AN ERROR
The sum of two numbers is nine. If the square of one number is three less than the other number, what are the numbers?
Let "x" equal the first number.
Let "y" equal the second number.
x + y = 9, and thus: x = 9 - y
x² = y - 3
Substitute (9 - y) for "x" in the above equation:
(9 - y)² = y - 3
9² - y² = y - 3
81 - y² = y - 3
y² + y - 3 - 81
y² + y - 84
Using the above for the quadratic equation, with: a = 1, b = 1, c = -84
y = [-b ± √(b² - 4ac)] / (2a)
y = [-1 ± √(1 - 4(1)(-84))] / [2(1)]
y = (-1 ± 18.36) / (2)
1st iteration of answer:
y = (-1 + 18.36) / (2), y = 8.68
x = 9 - y, x = 0.32
2nd iteration of answer:
y = (-1 - 18.36) / (2), y = -9.68
x = 9 - y, x = 18.68
STEPS TO CORRECT ANSWER
Let "x" equal the first number.
Let "y" equal the second number.
x + y = 9, and thus: y = 9 - x
x² = y - 3
Substitute (9 - x) for "y" in the above equation:
x² = 9 - x - 3
x² + x - 6
Using the above for the quadratic equation, with: a = 1, b = -1, c = -6
x = [-b ± √(b² - 4ac)] / (2a)
x = [-1 ± √(1 - 4(1)(-6))] / [2(1)]
x = (-1 ± 5) / (2)
1st iteration of answer:
x = (-1 + 5) / (2), x = 2
y = 9 - x, y = 7
2nd iteration of answer:
x = (-1 - 5) / (2), x = -3
y = 9 - x, y = 7
