please-help-There-is-21x-2-21p-x-49p-7-0-whose-roots-u-and-v-If-u-and-v-are-not-Z-and-u-v-1-find-the-value-of-u-v-

Question Number 49186 by afachri last updated on 04/Dec/18

please help \dots

There is :

21 x^{2} - 21 p x + 49 p - 7 = 0

whose roots u and v . If u and v are not \in Z,

and u, v ⩾ 1 .

find the value of u + v!

Commented by MJS last updated on 04/Dec/18

strange question \dots {there}^{'} s not a single solution

Commented by afachri last updated on 04/Dec/18

i thought it is . i found this question in test of University .

Commented by mr W last updated on 04/Dec/18

x = \frac{21 p \pm \sqrt{D}}{2 \times 21}

c o n d i t i o n 1 :

D = {(21 p)}^{2} - 4 \times 21 \times (49 p - 7) ⩾ 0

3 p^{2} - 4 (7 p - 1) ⩾ 0

3 p^{2} - 28 p + 4 ⩾ 0

p ⩽ \frac{14 - 2 \sqrt{46}}{3} = 0.145

p ⩾ \frac{14 + 2 \sqrt{46}}{3} = 9.188

c o n d i t i o n 2 :

x_{1} = \frac{21 p - \sqrt{D}}{2 \times 21} ⩾ 1

21 p - \sqrt{D} ⩾ 42

\sqrt{D} ⩽ 21 p - 42

D = {(21 p)}^{2} - 4 \times 21 \times (49 p - 7) ⩽ {(21 p - 42)}^{2}

1 ⩽ p

u + v = \frac{21 p}{21} = p ⩾ 2

u v = \frac{49 p - 7}{21} = \frac{7 p - 1}{3} ⩾ 1 \Rightarrow p ⩾ \frac{4}{7}

\Rightarrow u + v = p ⩾ \frac{14 + 2 \sqrt{46}}{3} = 9.188

Commented by afachri last updated on 04/Dec/18

nothing matches to the avaible answers . the options are

a . \frac{59}{3} b . \frac{60}{3} c . \frac{61}{3} d . \frac{62}{3} e . \frac{63}{3}

Answered by MJS last updated on 04/Dec/18

x^{2} - p x + \frac{7 p - 1}{3} = 0

x = \frac{p}{2} \pm \sqrt{\frac{3 p^{2} - 28 p + 4}{12}} \Rightarrow p ⩽ \frac{14 - 2 \sqrt{46}}{3} \lor p ⩾ \frac{14 + 2 \sqrt{46}}{3}

u + v = p with p ⩾ \frac{14 + 2 \sqrt{46}}{3} \land p \neq 13

Facebook Tweet Pin