Short Answer 
The Supreme Court’s unanimous 9-0 decision in Brown vs. Board ruled that segregated schools violated the Equal Protection Clause of the 14th Amendment, declaring segregation unconstitutional. This landmark ruling emphasized the need for equality in education, rejecting the flawed doctrine of “separate but equal” and setting the stage for school integration in the United States.
Step 1: Unanimous Decision
The Supreme Court reached a unanimous decision in the Brown vs. Board case, with all justices voting in favor of the plaintiffs. This means they agreed completely that the situation of segregation was unjust. The final vote count was 9-0, which indicates a strong consensus among the justices.
Step 2: Violation of Equal Protection
The court ruled that segregated schools violated the principle of equal protection under the laws. According to the justices, the longstanding doctrine of “separate but equal” contradicted the Equal Protection Clause of the 14th Amendment, which guarantees that all citizens receive equal treatment regardless of race. This ruling emphasized the necessity for equality in educational opportunities.
Step 3: Segregation Declared Unconstitutional
The court ultimately found that segregation itself was unconstitutional. The justices concluded that the doctrine of “separate but equal” was fundamentally flawed, asserting that such facilities were inherently unequal. This landmark decision paved the way for the integration of schools across America and reinforced the legal standpoint against racial segregation.