For the first question, you actually have to look up the formula for finding cubic roots.
Here's the thing: let the three roots be χ1, χ2 and χ3, then it's required that χ1:χ2=χ2:χ3. Common ratio right lol?
Hey, my neighbour again send me one exercise.
1. In the equation you must find solutions whose are in geometric sequence. For which ?
How can I find this ?
2. In the equation must find this that solutions are in arithmetic sequence.
How can I find this ?
For the first question, you actually have to look up the formula for finding cubic roots.
Here's the thing: let the three roots be χ1, χ2 and χ3, then it's required that χ1:χ2=χ2:χ3. Common ratio right lol?
For the second question, Δ=(a+3)^2-4(a+2)=a^2+6a+9-4a-8=a^2+2a+1=(a+1)^2
First, x^2=(1/2)[(a+3)+/-(a+1)] = (1/2)[(2a+4) or 2]= a+2 or 1.
From x^2= 1 we know x=-1 or 1.
Then there are 2 possibilities:
(1) 0<a+2<1 then -1< -sqrt(a+2) < sqrt(a+2) < 1, in which case, you have -sqrt(a+2)-(-1)=sqrt(a+2)-[-sqrt(a+2)].
(2) a+2>1 then -sqrt(a+2) < -1 < 1 < sqrt(a+2).
No, we don't need the general formula, here. Suppose our roots are $r_1, r_2,r_3$ in geometric sequence.
Then $r_2 = cr_1$, and $r_3 = cr_2 = c^2r_1$. Let us just use $r$ for $r_1$. So our polynomial factors as:
$x^3 - 7x - 21x + a = (x - r)(x - cr)(x - c^2r) = x^3 - (r + cr + c^2r)x^2 + (cr^2 + c^2r^2 + c^3r^2)x - c^3r^3$.
Evidently, then:
$r(1 + c + c^2) = 7$
$cr^2(1 + c + c^2) = -21$
$a = -c^3r^3$
From this we see that $cr = -3$. This is all we need to find $a$, since $a = -c^3r^3 = -(cr)^3 = -(-27) = 27$.
But we can do more:
Since $7 = r(1+c+c^2) = \dfrac{-3}{c}(1+c+c^2) = -\dfrac{3}{c} - 3 -3c$,
multiplying by $c$, we get:
$-3c - 3 - 3c^2 = 7 \implies c^2 + \dfrac{10}{3}c + 1 = 0$.
From the quadratic formula, we obtain:
$c = \dfrac{-\frac{10}{3} \pm \sqrt{\frac{100}{9} - 4}}{2} = -\dfrac{10}{6} \pm \dfrac{\sqrt{\frac{64}{9}}}{2} = -\dfrac{5}{3} \pm \dfrac{4}{3}$
so $c = -3$ or $c = -\dfrac{1}{3}$.
If $c = -3$ then since $cr = -3$ we must have $r = 1$. This works, because $1(1 - 3 + 9) = 1(7) = 7$. Thus our roots are $1,-3,9$.
If $c = -\dfrac{1}{3}$ then $r = 9$. This means our roots are $9,-3,1$, which are the same set of roots.
It is easy to see that:
$(x - 1)(x + 3)(x - 9) = x^3 - 7x^2 - 21x + 27$, as required.