Real have in past seasons found solace in the Champions League when struggling domestically, but this time found no respite from their current woes as Ajax produced a pulsating display of attacking football.
Thirteen-times winners Madrid, who have lifted the trophy in four of the last five years, and had reached at least the semi-finals for the last eight seasons, suffered their earliest elimination since 2010.
Madrid president Florentino Perez is now faced with deciding whether it is worth keeping Solari until the end of the season, let alone beyond the summer.
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Ajax though weren't in the mood to let the Spaniards back in the game and within two minutes they had put the gap back to two goals as Thibaut Courtois was in a awful position as Lasse Schone's free kick floated over his head and into the goal.
Sergio Ramos was in the stands, having earned a deliberate booking to clear his ban, believing the tie was already won.
The night could have gone very differently had Madrid defender Raphael Varane headed under the crossbar instead of against it in the fifth minute, moments before Ziyech scored.
It could have been far more uncomfortable, had VAR not ruled out a Nicolas Tagliafico goal in the first leg, or Kasper Dolberg not spurned a late chance when sent through on Thibaut Courtois's goal.
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They worked it from right to left and then inside again, where Donny van de Beek shifted one last pass to Tadic, who whipped a sizzling shot into the top corner.
TNT will broadcast Real Madrid versus Ajax in English, and UniMas will broadcast it in Spanish in the United States.
Asensio was able to pull one back for Real on 70 minutes, but Ajax's Lasse Schone cancelled that out soon after with an incredible free kick from the left side of Madrid's area to restore their three-goal lead to secure the historic win.
Ajax started with an unchanged side from the opening leg, with Frenkie de Jong set to play at the Santiago Bernabeu for the first time.
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