WEBVTT
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Factorise fully one minus 𝑥 squared plus 14𝑥𝑦 minus 49𝑦 squared.
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Our first step is to split the expression into two parts.
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If we factorise out negative one from the second, third, and fourth term, we are left with one minus 𝑥 squared minus 14𝑥𝑦 plus 49𝑦 squared.
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The second part of our expression can be factorised into two brackets, or parentheses, 𝑥 minus seven 𝑦 and 𝑥 minus seven 𝑦.
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We can check this by expanding the two parentheses using the F.O.I.L. method.
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Multiplying the first terms gives us 𝑥 squared, as 𝑥 multiplied by 𝑥 is equal to 𝑥 squared.
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Multiplying the outside terms gives us negative seven 𝑥𝑦.
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Multiplying the inside terms also gives us negative seven 𝑥𝑦.
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Finally, multiplying the last terms gives us 49𝑦 squared.
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Multiplying negative seven 𝑦 by negative seven 𝑦 gives 49𝑦 squared.
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This simplifies to 𝑥 squared minus 14𝑥𝑦 plus 49𝑦 squared.
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Therefore, our factorisation is correct.
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As the two parentheses are identical, this can be rewritten as one minus 𝑥 minus seven 𝑦 all squared.
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We now have an expression which is the difference of two squares, as one is a square number and 𝑥 minus seven 𝑦 all squared is also a square number.
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The difference of any two squares, 𝑎 squared minus 𝑏 squared, can be factorised to give 𝑎 minus 𝑏 multiplied by 𝑎 plus 𝑏.
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In our question, 𝑎 is equal to one and 𝑏 is equal to 𝑥 minus seven 𝑦.
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Our two parentheses, therefore, become one minus 𝑥 minus seven 𝑦 and one plus 𝑥 minus seven 𝑦.
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The first parenthesis simplifies to one minus 𝑥 plus seven 𝑦, as subtracting negative seven 𝑦 gives positive seven 𝑦.
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The second parenthesis simplifies to one plus 𝑥 minus seven 𝑦.
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This means that our fully factorised expression is one minus 𝑥 plus seven 𝑦 multiplied by one plus 𝑥 minus seven 𝑦.
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We can check this answer by expanding the two parentheses using the grid method.
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We have to ensure that we multiply every term in the first parenthesis by every term in the second parenthesis.
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One multiplied by one is equal to one.
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One multiplied by 𝑥 is equal to 𝑥.
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And one multiplied by negative seven 𝑦 is equal to negative seven 𝑦.
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Multiplying one, 𝑥, and negative seven 𝑦 by negative 𝑥 gives us negative 𝑥, negative 𝑥 squared, and positive seven 𝑥𝑦, respectively.
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Finally, multiplying one, 𝑥, and negative seven 𝑦 by seven 𝑦 gives us seven 𝑦, seven 𝑥𝑦, and negative 49𝑦 squared.
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The two 𝑥 terms and the two seven 𝑦 terms cancel.
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We are, therefore, left with one, negative 𝑥 squared, 14𝑥𝑦, and finally negative 49𝑦 squared.
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As this is equivalent to our initial expression, our factorisation is correct.