Worksheet: Trial and Improvement

In this worksheet, we will practice estimating the answers to some equations that do not have exact solutions using the method of trial and improvement.

Q1:

Daniel wants to find the points where the functions 𝑓(𝑥)=𝑥+8𝑥 and 𝑔(𝑥)=𝑥+100 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.

  • Ayes
  • Bno

Q2:

William is trying to find the solution to the equation 𝑥+8𝑥=100. He decides to use trial and improvement and has so far completed the table.

𝑥454.14.094.074.075
𝑥+8𝑥96165101.72

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

  • A110.43, 109.89, and 110.03
  • B49.45, 49.12, and 49.21
  • C44.99, 44.77, and 44.82
  • D101.14, 99.98, and 100.27
  • E156.85, 141.42, and 145.17

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=1,000 to two decimal places, given that it lies between 31 and 32.

Q4:

Use trial and improvement to solve the equation 𝑥+2𝑥=110, 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=4𝑦+20 to two decimal places, given that 𝑦 lies between 4 and 5.

Q6:

Use a trial and improvement method to solve the equation 𝑥+2𝑥25=0 to one decimal place, knowing that the equation has a solution between 2 and 3.

Q7:

The equation 𝑦3𝑦8=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+4𝑦=25 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 2𝑥5𝑥+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 104𝑥3=27, given that it lies between 1 and 2. Give your answer to two decimal places.

Q12:

Using the method of systematic trial and improvement, and giving your answer correct to one decimal place, find the solution between 𝑥=0 and 𝑥=1 and 2𝑥5𝑥+1=0.

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