Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically

step 1 It is given that the equation modulus of x minus 15 is equal to be x squared minus 15x. Now the

Solving an Equation Involving an Absolute Value Find all...

Key Concepts

Quadratic Equations

In solving equations that result from one or both cases of a piecewise analysis, it is common to encounter quadratic equations. Solving quadratic equations involves techniques such as factoring, completing the square, or applying the quadratic formula. Understanding these methods is essential for finding potential solutions and progressing through the problem.

Solution Verification

After obtaining potential solutions from the different cases, it is critical to substitute them back into the original equation to verify their validity. This step ensures that no extraneous solutions, which may arise from the process of splitting into cases or squaring both sides of an equation, are included in the final answer.

Absolute Value Function

The absolute value function, |x|, represents the distance of a number x from zero on the number line. It is defined as x when x is nonnegative and -x when x is negative. This concept is fundamental when dealing with equations that involve absolute values, as it guides the process of analyzing different cases based on the sign of the expression within the absolute value.

Piecewise Definition

Because the absolute value function is defined differently depending on whether its argument is positive or negative, it is often necessary to split the equation into two separate cases. Each case represents one piece of the piecewise definition and must be solved independently before combining the results, ensuring that the solutions adhere to the conditions assumed in each case.

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