B737-800 CAPT LOST DIRECTIONAL CTL BECAUSE BOTH ENGS WERE NOT ALLOWED TO STABILIZE AT PARTIAL PWR BEFORE TOGA WAS ENGAGED.

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TKOF ROLL. RWY 36R CVG. PWR ADVANCED TOWARDS TKOF THRUST. AUTOTHROTTLES COMMANDED TO ENGAGE. THROTTLES MOVING TOWARDS TKOF PWR SIMULTANEOUSLY. ACFT VEERED HARD L ON TKOF ROLL FROM ORIGINAL CTRLINE ALIGNMENT. NOSEWHEEL STEERING COMMANDED TOWARDS R TO COUNTERACT AGGRESSIVE PULL TOWARDS L. NO RESPONSE FROM NOSEWHEEL STEERING USING TILLER. APPLIED DIFFERENTIAL BRAKING TO CORRECT ACFT BACK TO CTRLINE. HOPPING/SCRAPING OF NOSEWHEEL OR DRAGGING OF L MAIN BRAKE CREATED BURNT RUBBER SMELL FROM SKIDDING TIRES. TKOF ROLL DISCONTINUED AT 15 KTS. ACFT WITHIN 30 FT FROM DEPARTING PREPARED SURFACE. CLRED ACTIVE RWY ONTO FIRST AVAILABLE TXWY. TOWED BACK TO GATE. HUMAN PERFORMANCE QUESTIONS: 1) NOSEWHEEL STEERING CTRED BEFORE TKOF ROLL? YES; I BELIEVE I ROLLED ACFT FORWARD ON TKOF ALIGNMENT. 2) DIFFERENTIAL THRUST APPLIED ON TKOF ROLL? I DON'T THINK SO. I DID USE #2 ENG PWR MORE DURING L TURN DURING ALIGNMENT ON POS AND HOLD; BUT I NOTED BOTH THROTTLES MOVING FORWARD TOWARD TKOF THRUST ON INITIAL PWR-UP.