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MATHS
Asked by Emil

The root of the quadratic equation 3 (power2) - 5x + p =0 is three times the value of other roo . Calculate the value of p Add Maths-Quadratic Equation?

There are 2 ways to solve this question. The first is covered by another person in the comments which is just equating the quadratic equation of the 2 roots and finding p. The second solution, in my opinion, is much more elegant and simple and requires less complex algebraic manipulation. This method works by setting our original equation to equal what it’s roots would be ie: 3x^2 -5x +p = 3(x-a)(x-b) (The 3 on the outside comes from our 3 as our 2nd power coefficient) Where a and b are roots to the equation. We know that a root is 3 times the other root so let’s set b = 3a =>> 3x^2 -5x +p = 3(x-a)(x-3a) Now let’s expand the right : => 3(x^2 - 4ax + 3a^2) => 3x^2 -12ax + 9a^2 Now we can compare coefficients of x so -12a = -5 9a^2 = p Therefore a=5/12 and p = 9(5/12)^2 p= 25/16

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Renat Samur
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The roots of a quadratic equation can be found with the quadratic formula: x = (-b ± sqrt(b^2-4ac) )/2a Putting our values of a, b and c into this formula we get x = (5±sqrt(25-12p)) / 6 Since one root is 3 times the other, we can set up the following equation: (5+sqrt(25-12p))/6 = 3*(5-sqrt(25-12p))/6 Solving this equation gives p = 25/16 To check, we can put this value of p back into our solution for the roots. This gives the roots as 5/4 and 5/12, so we are correct.

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Leo Penrose
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