Question

n² - 2n+28= 0 solve it by spliting the middle term plzzz answer fast plzzzz give the value of n... if not by splitting the middle term then try by other method

I am waiting and plz give answer in steps plzzzzzz​ any one give me the answer.

I am waiting and plz give answer in steps plzzzzzz​ any one give me the answer. plz give me the value of n​

Answer 1

Answer: 1±6√3i

Given:

$\sf n^2-2n+28$

To find:

$\sf Roots\:of\:the\:equation.$

Formula used:

$\sf For\:any\:quadratic\:equation\:like,ax^2+bx+c,$

$\boxed{\sf Discriminant\:of\:the\:equation(D):{b^2-4ac}}$

$\sf If,\\D>0\:\rightarrow Two\:real\:solutions\:exist\\D=0\:\rightarrow One\:real\:solution\:exist\\D<0\:\rightarrow No\:real\:solution\:exist(i.e.\:Two\:imaginary\:solutions\:exist)$

$\sf The\:solution\:for\:an^2+bn+c=0\:can\:be\:given\:by\:quadratic\:formula:$

$\boxed{\sf{n=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}}}$

Explanation:

$\sf According\:to\:the\:question,$

$\sf a=1\\b=-2\\c=28$

Finding D:

$\Rightarrow{2^2-4\times1\times28}$

$\Rightarrow \sf{4-112}$

$\Rightarrow \sf {-108}$

$\sf which\:means\:that,\:two\:imaginary\:solutions\:exist.$

$\sf On\:applying\:quadratic\:formula, we\:get:$

$\sf{n=\dfrac{2 \pm \sqrt{-108}}{2}}}$

$\sf \sqrt{-108}\:can\:be\:simplified\:as:$

$\Rightarrow \sf \sqrt{108\times-1}$

$\Rightarrow \sf \sqrt{108}\times\sqrt{-1}$

$\sf We\:know\:that\:value\:of\:i=\sqrt{-1}.Therefore,$

$\Rightarrow \sf \sqrt{108}i$

$\sf 108\:can\:be\:futher\:simplified\:as,$

$\Rightarrow\sf 6\sqrt{3}$

$\sf Hence,$

$\sf {\dfrac{2 + \sqrt{-108}}{2}}}=\sf{1 + \sqrt{108}i}=1+6\sqrt{3}i$

$\sf {\dfrac{2 - \sqrt{-108}}{2}}}=\sf{1 - \sqrt{108}i}=1-6\sqrt{3}i$

Therefore, the roots of the given quadratic equations are (1+6√3i) and (1-6√3i) respectively.