Moving at a constant speed along a Bézier Curve

Game Mechanics
#1

I’ve been playing with quadratic bezier curves recently, its easy enough to draw and get the points along it using something like:

float x = (1 - t) * (1 - t) * startX + 2 * (1 - t) * t * cx + t * t * endX;
float y = (1 - t) * (1 - t) * startY + 2 * (1 - t) * t * cy + t * t * endY;

Where start/end is the beginning/end point on the curve and c being the control point.

t is a value between 0 and 1 which can be used to move along the curve, 0 being the start and 1 being the end, however the problem is that this value can’t be used to move along the curve at a constant speed since t=0.5 is not necessarily at the middle of the curve or t=0.25 the first quarter.

I’ve got an approximation of the length of the curve using

public float getApproxLength() {
		float t = 0;
		length = 0; // reset length
		
		float oldX = startX;
		float oldY = startY;
		
		// scroll through curve and update length
		for (int i = 0; i < 100; i++) {
			t += 0.01f;
			
			float x = (1 - t) * (1 - t) * startX + 2 * (1 - t) * t * cx1 + t * t * endX;
			float y = (1 - t) * (1 - t) * startY + 2 * (1 - t) * t * cy1 + t * t * endY;
			
			length += getLineLength(x, y, oldX, oldY);
			
			oldX = x;
			oldY = y;
		}
		
		return length;
	}

Are there any easy solutions to achieve this other than to use a whole ton of look up tables and memory? doesn’t need to be super accurate just something good enough to get a decent constant looking movement along the curve.