Solve for x |x-10|=x^2-10x

Problem

|x−10|=x^2−10*x

Solution

1. Identify the definition of the absolute value, which requires considering two cases based on the sign of the expression inside the bars: x−10≥0 and x−10<0

2. Analyze Case 1 where x−10≥0 which means x≥10 In this case, |x−10|=x−10

x−10=x^2−10*x

0=x^2−11*x+10

(x−10)*(x−1)=0

x=10,x=1

3. Check the validity of Case 1 solutions against the condition x≥10 Only x=10 is valid.

4. Analyze Case 2 where x−10<0 which means x<10 In this case, |x−10|=−(x−10)=10−x

10−x=x^2−10*x

0=x^2−9*x−10

(x−10)*(x+1)=0

x=10,x=−1

5. Check the validity of Case 2 solutions against the condition x<10 Only x=−1 is valid.

6. Combine the valid solutions from both cases.

Final Answer

x=10,−1

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