# 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.

Key stage:  KS 4

Curriculum topic:  Algebra

Curriculum subtopic:  Use Algebra to Support and Construct Arguments

Difficulty level:

### 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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