A more thorough explanation:

Regents of the University of California v. Bakke is a 1978 Supreme Court case that dealt with a university's use of race in its admissions criteria. The Court held that using race as a definite and exclusive basis for an admission decision violated the Equal Protection Clause of the Fourteenth Amendment and Title VI of the Civil Rights Act of 1964.

For example, Bakke was a white male who applied to medical school at the University California at Davis. Although his admissions score was well above that of the average admittee and the school had open slots when he applied, his application was rejected because of the school's racial quota system. Previously, the school implemented a quota system where white applicants could only compete for 84 out of 100 spots, and the remaining 16 were reserved for racial minorities. Bakke sued the school, arguing that the clear-cut racial quota system was unconstitutional and a violation of the Civil Rights Act of 1964.

The Court held that these admission criteria violated the Equal Protection Clause and the Civil Rights Act of 1964. The Court applied strict scrutiny, reasoning that the Equal Protection Clause requires that a government have a compelling interest with narrowly tailored means to blatantly base their actions on race alone, as was the case here. In so reasoning, the court extends the heightened scrutiny on race discrimination to all races and does not limit it to the protection of racial minorities.

The medical school argued that their racial quota system served a compelling government interest by remedying the traditional underrepresentation of minorities in the medical profession and that the diversity in the classroom enhances the free exchange of ideas. The Court rejected the University's arguments though, finding that there are other ways to achieve representation of traditionally underrepresented groups and classroom diversity without a blatant racial quota system.

In summary, the case established that a quota system which excludes candidates because of their race alone is racial discrimination and that the University did not have a compelling reason with reasonably tailored means to overcome the constitutional standard of strict scrutiny.