# 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 and 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 . He decides to use trial and improvement and has so far completed the table.

4 | 5 | 4.1 | 4.09 | 4.07 | 4.075 | |

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

**Q4: **

Use trial and improvement to solve the equation , 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 to two decimal places, given that lies between 4 and 5.

**Q6: **

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

**Q7: **

The equation 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 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 between and . 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 , given that it lies between 1 and 2. Give your answer to two decimal places.