Question 38 of 43advanced🔧 ApplyShort Answer3 marks

If a number plus its reciprocal equals 10/3, find the number.

Correct Answer

3 or 1/3

Exercise: END-OF-CHAPTER EXERCISES | Q: 8 | (Page 23)
For More Understanding

Explanation

The problem requires setting up a quadratic equation from the given condition. By letting the number be x, we form the equation x + 1/x = 10/3. Multiplying by 3x yields 3x23x^{2} - 10x + 3 = 0. Factoring this quadratic equation using the splitting the middle term method (as demonstrated in the context for factoring expressions) gives the two possible values for the number.

Solution Steps

  1. Step 1: Let the number be x. The equation is x + 1/x = 10/3.

  2. Step 2: Multiply by 3x to get 3x23x^{2} + 3 = 10x, which simplifies to 3x23x^{2} - 10x + 3 = 0.

  3. Step 3: Find a and b such that a + b = -10 and ab = 9. The numbers are -1 and -9.

  4. Step 4: Factor by splitting the middle term: 3x23x^{2} - 9x - x + 3 = 0, which becomes (3x - 1)(x - 3) = 0.

  5. Step 5: Solve for x to get x = 3 or x = 1/3.

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