(x^(2) x-30)/(x-5) =1
How to Solve the Equation (\frac{x^2 \cdot x - 30}{x - 5} = 1)
Answer: Let’s solve the equation step by step.
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Simplify the Expression:
The expression (\frac{x^2 \cdot x - 30}{x - 5}) should first be simplified. Note that the expression (x^2 \cdot x) is simply (x^3). Thus, the expression becomes:
$$\frac{x^3 - 30}{x - 5} = 1$$
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Clear the Fraction:
To eliminate the fraction, multiply both sides of the equation by (x - 5) (assuming (x \neq 5) because division by zero is undefined):
$$(x^3 - 30) = 1 \cdot (x - 5)$$
Simplifies to:
$$x^3 - 30 = x - 5$$
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Rearrange the Equation:
Move all the terms to one side to set the equation to zero:
$$x^3 - x - 25 = 0$$
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Finding the Roots:
To solve (x^3 - x - 25 = 0), you may need to use methods such as synthetic division, factoring (if possible), or numerical methods for cubic equations if it cannot be factored easily.
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Check for Obvious Solutions:
You can test simple values like (x = 5) to see if they solve the equation. Remember, (x = 5) is not allowed in the original equation, but it might help simplify or factor further.
If you substitute (x = 3) as a guess:
$$3^3 - 3 - 25 \neq 0$$
You could continue testing values or use numerical approximations or graphing to find roots.
Summary: Solve the equation (\frac{x^3 - 30}{x - 5} = 1) by clearing the fraction, combining and rearranging terms, and then finding the roots of the resulting polynomial equation (x^3 - x - 25 = 0) using various algebraic methods.