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Given that πΏ and π are the roots of the equation two π₯ squared minus 21π₯ plus four equals zero, find, in its simplest form, the quadratic equation whose roots are two πΏ and two π.
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We begin by recalling the properties of a quadratic equation written in the form ππ₯ squared plus ππ₯ plus π equals zero, where π, π, and π are constants and π is nonzero.
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If this equation has roots π sub one and π sub two, then the sum of these is equal to negative π over π.
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The product of the two roots, π one multiplied by π two, is equal to π over π.
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In this question, weβre given the equation two π₯ squared minus 21π₯ plus four equals zero.
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The value of π is equal to two, π is equal to negative 21, and π is equal to four.
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We are told that the two roots are πΏ and π.
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Therefore, πΏ plus π is equal to negative negative 21 over two.
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This simplifies to 21 over two.
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The product of the roots πΏ multiplied by π is equal to four over two.
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This is equal to two.
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We have been asked to find the quadratic equation whose roots are two πΏ and two π.
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The sum of these roots is equal to two πΏ plus two π.
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These terms have a common factor of two, so this can be rewritten as two multiplied by πΏ plus π.
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We know that πΏ plus π is equal to 21 over two.
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Therefore, the sum of the roots equals two multiplied by 21 over two.
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This is equal to 21.
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Negative π over π is therefore equal to 21.
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The product of our two roots is equal to two πΏ multiplied by two π.
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This is equal to four πΏπ.
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As πΏπ is equal to two, we have four multiplied by two.
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We can therefore conclude that π over π is equal to eight.
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Using the two equations negative π over π equals 21 and π over π equals eight, we can calculate values for π, π, and π.
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This will in turn give us the quadratic equation whose roots are two πΏ and two π.
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As both 21 and eight are integers, we can let π equal one.
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This means that negative π is equal to 21 and π is therefore equal to negative 21.
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If π is equal to one from the second equation, π is equal to eight.
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The quadratic equation whose roots are two πΏ and two π is π₯ squared minus 21π₯ plus eight is equal to zero.