Well, well, well, let's find out! When a line is tangent to a parabola, that means it only touches the parabola at one point, like a fancy fingertip boop. So let's find the point where they make sweet, sweet contact.
We have the line equation y = -2x + b and the parabola equation y = 3x^2 + 4x - 1. To find the point of contact, we need the x-coordinate to be the same in both equations.
So, let's equate them:
-2x + b = 3x^2 + 4x - 1
Now, to solve this quadratic equation, let's rearrange it to the classic form:
3x^2 + 6x + (4x - 2 - b) = 0
To have only one solution for x, the discriminant (b^2 - 4ac) should be equal to zero. In this case, a = 3, b = 6, and c = (4x - 2 - b).
Let's plug the values into the quadratic formula, which is a bit like a math magician's hat:
b^2 - 4ac = 0
6^2 - 4(3)(4x - 2 - b) = 0
Simplifying things a bit:
36 - 12(4x - 2 - b) = 0
Now, let's do some beastly expanding:
36 - 48x + 24 - 12b = 0
Combining like terms:
60 - 48x - 12b = 0
To be a tangent, the line should touch the parabola at one point only. That means we have one solution for x, and that can only happen when the coefficient of x is zero:
-48x = 0
Solving this equation gives us:
x = 0
Now, let's substitute x = 0 into either the line or parabola equation. I'll choose the line equation to save time:
y = -2(0) + b
y = b
So when x = 0, y = b. That's the point of contact!
Therefore, the line y = -2x + b will be tangent to the parabola y = 3x^2 + 4x - 1 when b = 0. Ha! Zero is quite the star, isn't it?