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What happens to characteristics properties of quadratic equation if instead of x being real variable it is a complex variable properties like sum of roots& product of roots

What happens to characteristics properties of quadratic equation if instead of x being real variable it is a complex variable properties like sum of roots& product of roots

Grade:12

1 Answers

Ajay
209 Points
7 years ago
Posting again as previous answer did not get  posted correctly.
The properties of  roots of quadratic equation remains same even if the roots are not real
Let\quad f(x)\quad =\quad { x }^{ 2 }+2x+2\quad =\quad 0\\ x\quad =\quad -1+i,\quad -1-i\\ sum\quad of\quad roots\quad =\quad -2\\ Product\quad of\quad roots\quad =\quad (-1+i)(-1-i)\quad \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad 1-{ i }^{ 2 }\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad 2\\ There\quad are\quad some\quad intersting\quad consequences\quad of\quad if\quad a\quad quadratic\quad polynomial\quad \\ f(x)\quad ={ ax }^{ 2 }+bx+c\quad =\quad 0\quad have\quad imaginary\quad roots.\\ since\quad f(x)\quad \neq \quad 0\quad for\quad any\quad real\quad x\quad it\quad implies\quad that\quad the\quad graph\quad of\quad function\\ y=f(x)\quad =\quad { ax }^{ 2 }+bx+c\quad does\quad not\quad inetsect\quad x\quad axis\quad which\quad also\quad means\quad that\quad it\quad stays\quad \\ above\quad or\quad below\quad x\quad axis\quad for\quad all\quad x.Let\quad f(x)\quad =\quad { x }^{ 2 }+2x+2\quad =\quad 0\\ x\quad =\quad -1+i,\quad -1-i\\ sum\quad of\quad roots\quad =\quad -2\\ Product\quad of\quad roots\quad =\quad (-1+i)(-1-i)\quad =\quad 2\\ Infact\quad the\quad same\quad equation\quad can\quad be\quad written\quad as\\ { x }^{ 2 }-[(-1+i)+(-1-i)]x+(-1+i)(-1-i)\quad =\quad 0\\ \\ There\quad are\quad some\quad intersting\quad consequences\quad of\quad if\quad a\quad quadratic\quad polynomial\quad \\ f(x)\quad ={ ax }^{ 2 }+bx+c\quad =\quad 0\quad have\quad imaginary\quad roots.\\ since\quad f(x)\quad \neq \quad 0\quad for\quad any\quad real\quad x\quad it\quad implies\quad that\quad the\quad graph\quad of\quad function\\ y=f(x)\quad =\quad { ax }^{ 2 }+bx+c\quad does\quad not\quad inetsect\quad x\quad axis\quad which\quad also\quad means\quad that\quad it\quad stays\quad \\ above\quad or\quad below\quad x\quad axis\quad for\quad all\quad x.

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