Algebra Examples

Prove that a Root is on the Interval
,
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
The Intermediate Value Theorem states that, if is a real-valued continuous function on the interval , and is a number between and , then there is a contained in the interval such that .
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Calculate .
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Simplify each term.
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Remove parentheses around .
Raising to any positive power yields .
Multiply by to get .
Simplify by adding zeros.
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Add and to get .
Subtract from to get .
Calculate .
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Simplify each term.
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Remove parentheses around .
Raise to the power of to get .
Multiply by to get .
Simplify by adding and subtracting.
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Add and to get .
Subtract from to get .
Since is on the interval , solve the equation for at the root by setting to in .
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Rewrite the equation as .
The roots of this equation could not be found algebraically, so the roots were determined numerically.
The Intermediate Value Theorem states that there is a root on the interval because is a continuous function on .
The roots on the interval are located at .
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