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Find all integers n such that the quadratic 7x^2 + nx - 30  can be expressed as the product of two linear factors with integer coefficients.

 May 10, 2021
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Find all integers n such that the quadratic 7x^2 + nx - 30  can be expressed as the product of two linear factors with integer coefficients.

 

Hello Guest!

 

\(7x^2 + nx - 30 =0\)

 

\(x =\dfrac{-n\pm \sqrt{n^2+4\cdot 7\cdot 30}}{2\cdot 7}\\ x=-\dfrac{n}{14}\pm \dfrac{ \sqrt{n^2+840}}{14}\)

\(\{n^2+840)\}\subset \{square\ numbers \}\\ \{{\color{blue}11^2}+840)\}\subset \{square\ numbers \}\ determined\ graphically \\\)

\(x =\dfrac{-n\pm \sqrt{n^2+4\cdot 7\cdot 30}}{14}\\ x =\dfrac{-11\pm \sqrt{11^2+840}}{14}\\ x=\dfrac{-11\pm 31}{14}\\ \color{blue }x=-3\\ \)

\(\color{blue }7x^2 + 11x - 30 =0\\ 7\cdot (-3)^2 + 11\cdot (-3)- 30 =0\\\)

Help! I don't know what to do next.

sad  ! asinus

 May 11, 2021

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