Final answer:
To find the points of intersection between the two equations, we can solve the system of equations using the elimination method. After simplifying, we find that the points of intersection are (1, -2) and (3, 0).
Step-by-step explanation:
To find the points of intersection between the two equations, we need to solve the system of equations simultaneously. We can do this by either substitution or elimination method. Let's use the elimination method to solve these equations:
Multiply the first equation by 3 to make the coefficients of y in both equations match:
6x + 6y = -6
3y - 3x^2 = 12x - 9
Now, subtract the second equation from the first:
6x + 6y - (3y - 3x^2) = -6 - (-9)
6x + 6y - 3y + 3x^2 = 3
Combine like terms:
6x - 3x^2 + 6y - 3y = 3
Simplify:
3x^2 + 3x + 3y = 3
Divide the equation by 3:
x^2 + x + y = 1
Now we have the system of equations:
x^2 + x + y = 1
3y - 3x^2 = 12x - 9
Let's substitute the value of x from the first equation into the second equation:
3y - 3(1 - y) = 12(1 - y) - 9
Simplify:
3y - 3 + 3y = 12 - 12y - 9
Combine like terms:
6y - 3 = -12y + 3
Add 12y to both sides:
6y + 12y - 3 = 3
Combine like terms:
18y - 3 = 3
Add 3 to both sides:
18y = 6
Divide both sides by 18:
y = 6/18
Simplify:
y = 1/3
Now substitute the value of y back into the first equation:
x^2 + x + (1/3) = 1
Combine like terms:
x^2 + x + 1/3 = 1
Subtract 1/3 from both sides:
x^2 + x - 2/3 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the values from the equation: a = 1, b = 1, c = -2/3
Substitute these values into the quadratic formula:
x = (-1 ± √(1^2 - 4(1)(-2/3))) / (2(1))
Simplify:
x = (-1 ± √(1 + 8/3)) / 2
Combine like terms:
x = (-1 ± √(3/3 + 8/3)) / 2
Combine like terms:
x = (-1 ± √(11/3)) / 2
Simplify the square root:
x = (-1 ± √(11)/√(3)) / 2
Rationalize the denominator:
x = (-1 ± √(11)/√(3)) * √(3)/√(3) / 2
Simplify:
x = (-√(3) ± √(11))/2√(3)
So the points of intersection are:
(x, y) = (-√(3)/2√(3), 1/3) and (√(3)/2√(3), 1/3)
After simplifying, we get:
- (x, y) = (-1/2, 1/3)
- (x, y) = (1/2, 1/3)
Therefore, the correct answer is A) (1, -2) and (3, 0).