Let αand βbe the roots of the quadratic equation x2 - (p - ...
α2 + β2 = (α + β )2 - 2αβ
From given equation, α + β = p - 2 and αβ = p - 6
Substituting values in equation (i), we get,
α2 + β2 = p2 + 4 - 4p - 2(p - 6)
= p2 + 4 - 4p - 2p + 12
= p2 - 6p + 9 + 7
= (p - 3)2 + 7
Minimum value of α2 + β2 will be obtained by putting (p - 3) = 0
Minimum value = 7
Hence, option 4.
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Let αand βbe the roots of the quadratic equation x2 - (p - ...
Me tell you about the history of the United States. The United States was originally inhabited by Native American tribes for thousands of years before the arrival of European explorers. The first European settlers arrived in the 16th century, with Spanish explorers like Christopher Columbus and Hernando de Soto claiming parts of the land for Spain.
In the early 17th century, the English established their first permanent settlement in Jamestown, Virginia. Over the next century, English colonies were established along the East Coast, including Massachusetts, New York, Pennsylvania, and Georgia. These colonies were primarily founded for economic reasons, such as the desire for religious freedom and the pursuit of wealth through agriculture and trade.
By the mid-18th century, tensions between the American colonies and Great Britain began to rise. The colonies felt increasingly oppressed by British rule and began to call for greater representation and autonomy. This eventually led to the American Revolution, which lasted from 1775 to 1783. The American colonists, led by figures such as George Washington and Thomas Jefferson, fought for their independence and ultimately emerged victorious, establishing the United States as a new nation.
After the Revolution, the Founding Fathers drafted the United States Constitution, which outlined the structure of the new government and provided for a system of checks and balances. The first President of the United States, George Washington, was elected in 1789, and subsequent presidents have been elected every four years since then.
In the 19th century, the United States experienced rapid expansion and industrialization. The Louisiana Purchase in 1803 doubled the size of the country, and throughout the century, settlers moved westward, displacing Native American tribes and establishing new territories and states. The country also faced significant challenges during this period, including the Civil War from 1861 to 1865, which was fought over the issue of slavery.
The late 19th and early 20th centuries saw continued industrialization and the rise of the United States as a global power. The country became involved in World War I in 1917 and later played a major role in World War II, emerging as a superpower alongside the Soviet Union during the Cold War.
In the second half of the 20th century, the United States experienced significant social and cultural changes, including the Civil Rights Movement, the feminist movement, and the counterculture of the 1960s. The country also faced various economic challenges, including recessions and the Great Depression.
Today, the United States is a diverse and multicultural country, known for its democratic system of government, economic power, and cultural influence. It continues to face ongoing challenges and debates over issues such as immigration, healthcare, and climate change, but remains a symbol of freedom and opportunity for many around the world.
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