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Given that πΏ plus three and π plus three are the roots of the equation π₯ squared plus eight π₯ plus 12 is equal to zero, find in its simplest form the quadratic equation whose roots are πΏ and π.
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In this question, we are being asked to work backwards from knowing the roots of a quadratic equation to finding the quadratic equation itself.
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Letβs consider how to do this.
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If the quadratic equation has roots πΏ and π, then it can be factorized as π₯ minus πΏ multiplied by π₯ minus π.
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Now, letβs multiply out the brackets to see what the expanded form of the quadratic would look like in terms of πΏ and π.
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We would have π₯ squared minus πΏπ₯ minus ππ₯ plus πΏπ is equal to zero.
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The π₯ terms can be factorized giving π₯ squared minus πΏ plus ππ₯ plus πΏπ is equal to zero.
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Now, letβs look at the coefficient in its expanded form more closely.
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The coefficient of π₯ is negative πΏ plus π and πΏ plus π is the sum of the two roots.
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The constant term is plus πΏπ which is plus the product of the roots.
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What this means is that if we can find the sum of πΏ and π and the product of πΏ and π, then weβll be able to determine the quadratic equation whose roots they are.
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Weβve been told in the question about a different quadratic equation π₯ squared plus eight π₯ plus 12 is equal to zero, whose roots are πΏ plus three and π plus three.
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Letβs see how this helps.
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We can first express this quadratic slightly differently by writing the middle term positive eight π₯ as negative negative eight π₯.
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Using what weβve just determined, this means that for this quadratic, the sum of its roots is negative eight and the product of its roots is 12.
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The roots of this quadratic remember are πΏ plus three and π plus three.
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So we can form some equations.
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Firstly, as the sum of the roots is negative eight, we have the equation πΏ plus three plus π plus three is equal to negative eight.
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Secondly, as the product of the roots is 12, we have the equation πΏ plus three multiplied by π plus three is equal to 12.
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We can now use these two equations to find the sum and product of πΏ and π so that we can substitute them back into the equation of the quadratic weβre looking to find.
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The first equation simplifies by adding the two threes together to give πΏ plus π plus six is equal to negative eight.
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Subtracting six from both sides gives πΏ plus π is equal to negative 14.
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And now, weβve found the sum of πΏ and π.
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This means that weβll be able to determine the coefficient of π₯ in our quadratic equation.
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The sum of πΏ and π is negative 14, which means a coefficient of π₯ will be minus negative 14 or 14.
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Now, letβs consider the second equation.
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Expanding the double brackets gives πΏπ plus three πΏ plus three π plus nine is equal to 12.
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Subtracting nine from both sides of the equation gives πΏπ plus three πΏ plus three π is equal to three.
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Remember weβre still looking to determine the product of these roots β πΏπ.
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So this equation looks like itβs going to be useful.
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We can factorize plus three πΏ plus three π to give πΏπ plus three lots of πΏ plus π is equal to three.
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Remember weβve already determined what the sum of πΏ and π is.
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Itβs negative 14, which means we can substitute this value into the equation.
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We now have πΏπ plus three multiplied by negative 14 is equal to three.
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Three multiplied by negative 14 is negative 42.
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And if we add 42 to both sides, we find that πΏπ is equal to 45.
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So now, we also know the product of πΏ and π.
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And we can use this to find the constant term in our quadratic equation.
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Substituting the values we found for πΏ plus π and πΏπ into the quadratic equation, we have π₯ squared minus negative 14π₯ plus 45 is equal to zero.
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Simplifying the coefficient for the π₯ terms gives our quadratic equation in its simplest form.
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The quadratic equation whose roots are πΏ and π is π₯ squared plus 14π₯ plus 45 is equal to zero.