a) px^2 + 3x + q=0 has roots [((α+β)/(αβ))]× αβ find p and q if x^2 − 7x + 4 = 0 with real and distinct roots has same roots.. b)if α and β are roots of x^2 + kx +2k+8=0. a) find k if one root is twice the other.


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Question Number 33322 by Rio Mike last updated on 15/Apr/18

$a) {p x}^{2} + 3 x + q = 0 h a s r o o t s$ $[\frac{α + β}{α β}] \times α β$ $f i n d p a n d q i f$ $x^{2} - 7 x + 4 = 0 w i t h r e a l a n d$ $d i s t i n c t r o o t s h a s s a m e r o o t s . .$ $b) i f α a n d β a r e r o o t s o f$ $x^{2} + k x + 2 k + 8 = 0 .$ $a) f i n d k i f o n e r o o t i s t w i c e t h e o t h e r .$
Commented by Rio Mike last updated on 15/Apr/18

$y e s s i r t a k e i t t h e s e w a y i t r i e d s o l v i n g i t$ $x^{2} + k x + 2 k + 8 = 0$ $α + β = - k a n d α β = 2 k + 8$ $l e t o n e r o o t b e α a n d t h e o t h e r 2 α$ ${S O R}_{n} = α + 2 α = - k \Rightarrow \frac{3 α}{3} = \frac{- k}{3}$ $α = \frac{- k}{3}$ ${P O R}_{n} α (2 α) = 2 k + 8$ $2 α^{2} - 2 k - 8 = 0$ $2 {(\frac{- k}{3})}^{2} - 2 k - 8 = 0$ $k^{2} - 9 k - 36 = 0$ $k = 3 o r k = 12$
Commented by MJS last updated on 15/Apr/18

$x^{2} - 7 x + 4 = 0$ $x = \frac{7}{2} \pm \sqrt{\frac{49}{4} - 4} = \frac{7}{2} \pm \frac{\sqrt{33}}{2}$ $α = \frac{7 - \sqrt{33}}{2}$ $β = \frac{7 + \sqrt{33}}{2}$ ${p x}^{2} + 3 x + q = 0$ $x^{2} + \frac{3}{p} x + \frac{q}{p} = 0$ $\frac{3}{p} = - 7 \Rightarrow p = - \frac{3}{7}$ $\frac{q}{p} = 4 \Rightarrow q = - \frac{12}{7}$ $- \frac{3}{7} x^{2} + 3 x - \frac{12}{7} = 0$
Commented by MJS last updated on 15/Apr/18

$sorry somehow I thought it was$ $x^{3} . . . I might need new glasses; -)$ $anyway your answer is right$
Commented by Rasheed.Sindhi last updated on 16/Apr/18

$What is the use of [\frac{α + β}{α β}] \times α β ?$
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