KingoRoot is often considered the most successful tool for Lollipop devices. It uses a cloud-based database of exploits to find a path to root access.
Open source, highly secure, fine-grained permission management for individual apps, no bloatware.
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If one-click apps fail, you can use a second Android device to act as a "computer" to send commands to your target phone. Preparation : Install the Bugjaeger Mobile ADB Android phone. Connection : Connect both phones using a USB OTG (On-The-Go) cable.