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Part (a) Factorise 𝑥 squared plus six 𝑥 plus nine.
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This is a quadratic expression as the highest power of 𝑥 is two.
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To factorise this expression means to put it into brackets which we’ll multiply together to give the original quadratic.
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As the coefficient of 𝑥 squared — that means the number in front of 𝑥 squared — is just one, although we haven’t written it, then this means that the first term in each bracket is just 𝑥 as when we multiply 𝑥 by 𝑥, we get 𝑥 squared.
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We then need to work out what numbers go inside each bracket so that we get the original quadratic when we expand.
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When we have a quadratic of this form, we recall that the two numbers we’re looking for need to sum or add to the coefficient of 𝑥.
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So that’s six in this case.
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And they need to multiply to the constant term, which in this case is nine.
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We can find this pair of numbers by first listing the factors of nine.
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We have one and nine or three and three.
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Three and three also add to six.
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So these are the two numbers that we’re looking for.
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We, therefore, write plus three in each bracket.
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And our quadratic has been factorised.
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As our two brackets are identical, we have 𝑥 plus three multiplied by itself, 𝑥 plus three.
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We can also write this as 𝑥 plus three squared.
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We can give our answer in either of these two forms.
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Part (b) of the question says, “Solve eight 𝑦 minus two is greater than or equal to two 𝑦 plus 10.”
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Notice that here we’re being asked to solve an inequality.
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I read this sign as greater than or equal to.
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You can remember which way around the inequality sign go by remembering that the larger part of the sign is next to the largest side of the inequality or that the arrow points towards the smaller value.
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Solving an inequality is a lot like solving an equation.
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But instead of getting just one or perhaps two values for our answer, we instead get a range of values.
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Just like when we solve an equation, we need to make sure that we perform the same operation to each side of our inequality.
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We notice first that we have terms involving 𝑦 on each side.
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So our first step is going to be to collect them on the same side.
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It’s always easier to collect terms on the side which has the larger number of that variable to start with.
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So we’re going to collect on the left of the inequality.
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To do so, we need to subtract two 𝑦 from each side.
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On the left, we now have six 𝑦 because that’s eight 𝑦 minus two 𝑦 minus two.
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And on the right, we just have 10.
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So our inequality has become six 𝑦 minus two is greater than or equal to 10.
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Our next step is to add two to each side.
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This cancels the negative two on the left.
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And on the right, we now have 12.
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So we have six 𝑦 is greater than or equal to 12.
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To solve for 𝑦, our final step is to divide both sides by six as six 𝑦 divided by six gives 𝑦.
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On the right-hand side, 12 divided by six gives two.
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So the solution to our inequality is 𝑦 is greater than or equal to two.
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This includes all of the values from two upwards.
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So it would include integer values like five.
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But it would also include decimal values like 8.2 or even irrational numbers like 𝜋.
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Now, just one brief word about solving inequalities.
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I said that they were very similar to solving equations.
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And that’s true.
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But there is one key difference.
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And this occurs if we multiply or divide both sides of an inequality by a negative value.
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If we do this, then we must remember to reverse the inequality sign.
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We can see this if we think of the simple inequality two is less than 10.
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If we multiplied both sides of this inequality by negative one but didn’t change the direction of the sign, then we would have the statement negative two is less than negative 10.
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However, this is completely untrue.
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Negative two is actually greater than negative 10 because it’s less far below zero.
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So when we multiplied by negative one, we also needed to reverse the direction of the inequality.
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Although that wasn’t required in this question, it’s really important that you remember this when solving inequalities.