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Trial and Improvement to Find Approximate Solutions

In this worksheet, students use trial and improvement to home in on an approximate solution to an equation.

'Trial and Improvement to Find Approximate Solutions' worksheet

Key stage:  KS 4

Curriculum topic:  Algebra

Curriculum subtopic:  Use Algebra to Support and Construct Arguments

Difficulty level:  

down

Worksheet Overview

QUESTION 1 of 10

This worksheet is about finding approximate solutions to more complex equations using trial and improvement.

 

We test a value which is near the solution and then home in to a more accurate solution until we reach the desired level of accuracy. 

 

Example

One solution to the following equation lies between x = 3 and x = 4.

x3 + 2x2 + 3x = 100


Using trial and improvement, find this solution to 1 d.p.

 

 x  x3 + 2x2 + 3x  Comment
 3  33 + 2×32 + 3×3 = 54  too small
 4  43 + 2×42 + 3×4= 108  too big
 3.5  3.53 + 2×3.52 + 3×3.5= 77.875  too small
 3.8  3.83 + 2×3.82 + 3×3.8= 95.152  too small
 3.9  3.93 + 2×3.92 + 3×3.9= 101.439  just too big
 3.85  3.853 + 2×3.852 + 3×3.85= 98.261625  just too small

 

This shows that x = 3.85 is just too small but x = 3.9 is just too big, so the solutions lies between 3.85 and 3.9.

So to 1 decimal place, we can be sure that it will be x = 3.9 because anything above 3.85 will automatically round up to 3.9 anyway.

So x = 3.9 (to 1 d.p.)

One solution to the following equation lies between x = 3 and x = 4.

 

x3 + x = 50


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

One solution to the following equation lies between x = 4 and x = 5.

 

x3 + x2 + 3x = 100


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

One solution to the following equation lies between x = 2 and x = 3.

 

x3 + 2x2 + 3x = 50


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

One solution to the following equation lies between x = 3 and x = 4.

 

x3 + 3x2 - 4x = 50


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

One solution to the following equation lies between x = 5 and x = 6.

 

x3 - 4x2 + 10x = 100


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

One solution to the following equation lies between x = 8 and x = 9.

 

x3 - x2 - x = 500


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

One solution to the following equation lies between x = 7 and x = 8.

 

x3 + 2x2 - x = 500


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

One solution to the following equation lies between x = 6 and x = 7.

 

2x3 + 2x2 - x = 500


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

One solution to the following equation lies between x = 4 and x = 5.

 

2x3 + 2x2 + x = 200


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

One solution to the following equation lies between x = -3 and x = -4.

 

x3 + 3x2 + 4x = -25


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

  • Question 1

One solution to the following equation lies between x = 3 and x = 4.

 

x3 + x = 50


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
3.6
EDDIE SAYS
3.55³ + 3.55 = 48.288875 (just too small)
3.6³ + 3.6 = 50.256 (just too big)
  • Question 2

One solution to the following equation lies between x = 4 and x = 5.

 

x3 + x2 + 3x = 100


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
4.1
EDDIE SAYS
4.1³ + 4.1² + 3 × 4.1 = 98.031 (just too small)
4.15³ + 4.15² + 3 × 4.15 = 101.14588 (just too big)
  • Question 3

One solution to the following equation lies between x = 2 and x = 3.

 

x3 + 2x2 + 3x = 50


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
2.9
EDDIE SAYS
2.9³ + 2 × 2.9² + 3 × 2.9 = 49.909 (just too small)
2.95³ + 2 × 2.95² + 3 × 2.95 = 51.927375 (just too big)
  • Question 4

One solution to the following equation lies between x = 3 and x = 4.

 

x3 + 3x2 - 4x = 50


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
3.2
EDDIE SAYS
3.15³ + 3 × 3.15² - 4 × 3.15 = 48.423375 (just too small)
3.2³ + 3 × 3.2² - 4 × 3.2 = 50.688 (just too big)
  • Question 5

One solution to the following equation lies between x = 5 and x = 6.

 

x3 - 4x2 + 10x = 100


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
5.5
EDDIE SAYS
5.45³ - 4 × 5.45² + 10 × 5.45 = 97.568625 (just too small)
5.5³ - 4 × 5.5² + 10 × 5.5 = 100.375 (just too big)
  • Question 6

One solution to the following equation lies between x = 8 and x = 9.

 

x3 - x2 - x = 500


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
8.3
EDDIE SAYS
8.3³ - 8.3² - 8.3 = 494.597 (just too small)
8.35³ - 8.35² - 8.35 = 504.110375 (just too big)
  • Question 7

One solution to the following equation lies between x = 7 and x = 8.

 

x3 + 2x2 - x = 500


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
7.4
EDDIE SAYS
7.35³ + 2 × 7.35² - 7.35 = 497.760375 (just too small)
7.4³ + 2 × 7.4² - 7.4 = 507.344 (just too big)
  • Question 8

One solution to the following equation lies between x = 6 and x = 7.

 

2x3 + 2x2 - x = 500


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
6.0
EDDIE SAYS
2 × 6.0³ + 2 × 6.0² - 6.0 = 498 (just too small)
2 × 6.05³ + 2 × 6.05² - 6.05 = 510.04525 (just too big)
  • Question 9

One solution to the following equation lies between x = 4 and x = 5.

 

2x3 + 2x2 + x = 200


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
4.3
EDDIE SAYS
2 × 4.25³ + 2 × 4.25² + 4.25 = 193.90625 (just too small)
2 × 4.3³ + 2 × 4.3² + 4.3 = 200.294 (just too big)
  • Question 10

One solution to the following equation lies between x = -3 and x = -4.

 

x3 + 3x2 + 4x = -25


Using trial and improvement, find this solution to 1 d.p.

 

(just write the number)

CORRECT ANSWER
-3.7
EDDIE SAYS
(-3.75)³ + 3 × (-3.75)² + 4 × (-3.75) = -25.546875 (just too small)
(-3.7)³ + 3 × (-3.7)² + 4 × (-3.7) = -24.383 (just too big)
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