Question

Solve th equation -
$25 {x}^{2}- 30x + 11 = 0$
by using the general expression for the roots of a quadratic equation and showw that the roots are complex conjugate


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Answer 1

$Given \: quadratic \: equation : \\25x^{2} - 30x + 11 = 0$

$Compare \: this \: with \: ax^{2}+bx+c=0 ,we\:get$

$a = 25 , b = -30, c = 11$

$Discreminant (D) = b^{2} - 4ac \\= (-30)^{2} - 4 \times 25 \times 11 \\= 900 - 1100 \\= - 200$

$D < 0 \\\blue {( Roots \:are\: complex \: conjugate) }$

$\underline { \pink {Using \: Quadratic \: formula :}}$

$\boxed { \orange { x = \frac{-b±\sqrt{D}}{2a}}}$

$= \frac{-(-30)±\sqrt{-200}}{2\times 25}$

$= \frac{30±\sqrt{100\times 2 \times i^{2}}}{50}\\= \frac{30±10i\sqrt{2}}{50}\\= \frac{10(3±\sqrt{2}i)}{50}\\ = 3±\sqrt{2}i$

Therefore.,

$\green { complex \: conjugate \: roots \:are }$

$\green {(3+\sqrt{2}i),(3-\sqrt{2}i) }$

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