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| − | In Mathematics, '''Jadhav's Prime Quadratic Theorem''' is based on [[Algebra]] and [[Number Theory]]. Discovered by an Indian Mathematician [[Jyotiraditya Jadhav]]. Stating a condition over the value of <math>x</math> in the [[quadratic equation]] <math>ax^2+bx+c</math>.
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| − | == Theorem ==
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| − | It states that if a [[quadratic equation]] <math>ax^2+bx+c</math> is divided by <math>x</math> then it gives the answer as an [[Integer]] if and only if <math>x </math> is equal to 1, [[Integer_factorization|Prime Factors]] and [[composite]] [[divisor]] of the constant <math>c</math> .
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| − | <math>\frac{ax^2+bx+c}{x} \in Z </math> Iff <math>x</math> is a factor of <math>c</math> where <math>a,b,c \in Z </math>.
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| − | == Historical Note ==
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| − | [[Jyotiraditya Jadhav]] is a school student and is always curious about [https://www.ck12.org/book/ck-12-middle-school-math-concepts-grade-7/section/1.2 numerical patterns] which fall under the branch of [[Number Theory]]. He formulated many [https://en.wikipedia.org/wiki/Arithmetic arithmetic based equations] before too like [[Jadhav Theorem]], [[Jadhav Triads]], [[Jadhav Arithmetic Merging Equation]] and many more. While he was solving a question relating to [[quadratic equation]]s he found out this numerical pattern and organized the [[theorem]] over it.
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| − | == Proof ==
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| − | Now let us take <math>\frac{ax^2+bx+c}{x} </math> written as <math>\frac{x[ax+b]+c}{x} </math>
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| − | To cancel out <math>x </math> from the denominator we need <math>x </math> in numerator and to take <math>x </math> as common from whole quadratic equation we need to have <math>c </math> as a composite number made up as prime-factors with at least one factor as <math>x </math> or in other words <math>c </math> should be a multiple of <math>x </math> and hence telling us <math>x </math> should at least be a prime factor, composite divisor or 1 to give the answer as an Integer.
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| − | Hence Proving Jadhav Prime Quadratic Theorem.
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| − | '''Original Research paper''' can be found [https://issuu.com/jyotiraditya123/docs/jadhav_prime_quadratic_theorem here on Issuu]
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