To find the coordinates at which the parabola intersects the line y = 2x + 3, we need to set the equations of the parabola and the line equal to each other.
Let's assume the equation of the parabola is y = ax^2 + bx + c, where a, b, and c are constants.
Setting this equation equal to y = 2x + 3, we have:
ax^2 + bx + c = 2x + 3
Now, to solve for x, we need to set this equation equal to zero:
ax^2 + bx + c - 2x - 3 = 0
Combining like terms, we have:
ax^2 + (b - 2)x + (c - 3) = 0
Now, since this is a parabola, it can intersect the line at zero, one, or two points. To find the coordinates, let's consider the discriminant of this quadratic equation: b^2 - 4ac.
If the discriminant is positive, it means the quadratic equation has two distinct real roots and the parabola intersects the line at two points.
If the discriminant is zero, it means the quadratic equation has one real root (which is a double root) and the parabola intersects the line at one point.
If the discriminant is negative, it means the quadratic equation has no real roots and the parabola does not intersect the line.
Let's calculate the discriminant:
Discriminant = (b - 2)^2 - 4a(c - 3)
= b^2 - 4b + 4 - 4ac + 12a
= -3b + 4 - 4ac + 12a
Now, we have to consider three cases:
1) If the discriminant > 0, i.e., -3b + 4 - 4ac + 12a > 0, then the parabola intersects the line at two points.
2) If the discriminant = 0, i.e., -3b + 4 - 4ac + 12a = 0, then the parabola intersects the line at one point.
3) If the discriminant < 0, i.e., -3b + 4 - 4ac + 12a < 0, then the parabola does not intersect the line.
Without knowing the specific values of a, b, and c, we cannot determine the points at which the parabola intersects the line y = 2x + 3.
Find the coordinates at which the parabola intersects the line y = 2x + 3
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