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Trial & Improvement

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Lesson Aims To know how to use trial & improvement methods to solve equations.

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What is Trial & Improvement It’s a method where the unknown value in an equation can be found by first making a guess. The guess is improved step by step until the required degree of accuracy has been achieved. For example….

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52 cm 2 How did we find the area of this square? x x

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x 2 = 52 cm 2 We need to calculate x. Can you think of a number that when it’s squared is close to 52? 7 is too small (49) & 8 is too big(64) So the answer must be somewhere in between.

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Set up a table of values for x 2 = 52 x x 2 Comment 749Too small 864Too big 7.556.25Too big 7.353.29Too big 7.251.84Too small 7.2552.56Too big 7.24 52.4176 Too big

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Accuracy How accurate do we need our answer to be? Normally the question states the number of decimal places to work to, this is usually 1 or 2 dp

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Back to our example From the table we know that 7.2 was too small, 7.24 too big. Therefore our answer lies somewhere between them. If we want accuracy to 1 dp, then we look at the answer with two dp which will indicate whether we have to round up or down to get our final answer. Here we know that any number between 7.2 & 7.24 will be rounded to 7.2. therefore x=7.2 to 1dp. What would we do for an answer to accurate to 2 dp? 7.27.24

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Let’s find out (accuracy 2 dp) xx 2 Comment 7.24 52.4176Too Big 7.22 52.1284Too Big 7.21 51.9841Too Small 7.215 52.05623 Too Big 7.214 52.041796 Too Big Do we need to go any further?

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Let’s see We know the answer lies between 7.21 & 7.214 So, any number between these points will be rounded down to 7.21 to give an accuracy of 2 dp. 7.217.214

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Now it’s your turn! The answer to x 2 =15 lies somewhere between 3 & 4. Find x correct to 1 dp. The first thing you have to do is draw up a table of values & comments. Here you need: x x 2 Comments

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The question states that the answer lies between 3 & 4 so what will be your initial guess? 3.5 Off you go!

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How did you do? x x 2 Comment 3.512.25Too small 3.713.69Too small 3.814.44Too small 3.915.21Too big 3.8514.8225Too small

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The answer lies between 3.85 & 3.9 Any number between these will be rounded up to 3.9 to 1 dp 3.85 3.9

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