Worksheet: Approximating Roots by Trial and Improvement

In this worksheet, we will practice using systematic trial and improvement to find a root of a quadratic equation.

Q1:

Daniel wants to find the points where the functions 𝑓 ( 𝑥 ) = 𝑥 + 8 𝑥 and 𝑔 ( 𝑥 ) = 𝑥 + 1 0 0 are equal. He decides to use trial and improvement and finds that 7.10 is a solution to two decimal places. By sketching the graphs, determine whether he has found all of the solutions.

  • A yes
  • B no

Q2:

William is trying to find the solution to the equation 𝑥 + 8 𝑥 = 1 0 0 . He decides to use trial and improvement and has so far completed the table.

𝑥 4 5 4.1 4.09 4.07 4.075
𝑥 + 8 𝑥 96 165 101.72

Work out the three missing outputs from his table, giving your answer to two decimal places.

  • A156.85, 141.42, and 145.17
  • B 44.99, 44.77, and 44.82
  • C 49.45, 49.12, and 49.21
  • D 101.14, 99.98, and 100.27
  • E 110.43, 109.89, and 110.03

Hence, work out the solution to the equation to two decimal places.

Q3:

By using a trial and improvement method, find the positive solution of 𝑦 + 𝑦 = 1 , 0 0 0 to two decimal places, given that it lies between 31 and 32.

Q4:

Use trial and improvement to solve the equation 𝑥 + 2 𝑥 = 1 1 0 , given that the equation has a solution between 4 and 5. Approximate your answer to one decimal place.

Q5:

By using the trial and improvement method, find the solution of 𝑦 = 4 𝑦 + 2 0 to two decimal places, given that 𝑦 lies between 4 and 5.

Q6:

Use a trial and improvement method to solve the equation 𝑥 + 2 𝑥 2 5 = 0 to one decimal place, knowing that the equation has a solution between 2 and 3.

Q7:

The equation 𝑦 3 𝑦 8 = 0 has a solution between 2 and 3. Use trial and improvement to find this solution correct to one decimal place.

Q8:

Use a trial and improvement method to find the square root of 1,000. Give your answer to the nearest whole number.

Q9:

The equation 𝑦 + 4 𝑦 = 2 5 has a solution between 3 and 4. Use trial and improvement to find this solution. Give your answer correct to one decimal place.

Q10:

Find the solution of 2 𝑥 5 𝑥 + 1 = 0 between 𝑥 = 0 and 𝑥 = 1 . Use trial and improvement. Give your answer correct to one decimal place.

Q11:

Use the method of trial and improvement to find the positive solution of the equation 1 0 4 𝑥 3 = 2 7 , given that it lies between 1 and 2. Give your answer to two decimal places.

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