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Find the solution set of 𝑥 plus nine all squared is equal to 𝑥 plus nine in the set of real numbers.
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There are many ways of approaching this problem.
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We might notice by inspection that if we substitute 𝑥 equals negative nine into both sides of the equation, we get an answer of zero.
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Whilst this will be one solution to our equation, we need to ensure we have all the real solutions.
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We can do this by firstly expanding our parentheses.
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𝑥 plus nine all squared is equal to 𝑥 plus nine multiplied by 𝑥 plus nine.
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We can distribute the parentheses here by using the FOIL method.
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This gives us 𝑥 squared plus nine 𝑥 plus nine 𝑥 plus 81.
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Collecting like terms, this simplifies to 𝑥 squared plus 18𝑥 plus 81.
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We know this is equal to 𝑥 plus nine.
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Subtracting 𝑥 and subtracting nine from both sides gives us 𝑥 squared plus 17𝑥 plus 72 is equal to zero.
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This quadratic can be solved by factoring.
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As the coefficient of 𝑥 squared is equal to one, the first term in each of our parentheses will be 𝑥.
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We then need to find two integers that have a product of 72 and a sum of 17.
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Nine times eight is equal to 72, and nine plus eight is equal to 17.
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𝑥 squared plus 17𝑥 plus 72 is equal to 𝑥 plus nine multiplied by 𝑥 plus eight.
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As the product of our parentheses equals zero, either 𝑥 plus nine equals zero or 𝑥 plus eight equals zero.
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This gives us two solutions negative nine and negative eight.
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When 𝑥 is equal to negative eight, both sides of the equation equal one.
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And when 𝑥 is equal to negative nine, both sides of our equation equal zero.
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The solution set of 𝑥 plus nine all squared equals 𝑥 plus nine is negative eight and negative nine.