CoolBerth

The given equation is:
2x2+4x−1=x2−8x+3

Rearranging terms, we get:
2x2+4x−1−x2+8x−3=0
x2+12x−4=0
Now, the sum of the squares of the roots of this equation can be found using the formula:

Sum of the squares of the roots=(Sum of the roots)2−2(Product of the roots)
 

For the equation $x^2 + bx + c = 0$, the sum of the roots is $-b$ and the product of the roots is $c$. Therefore, for our equation $x^2 + 12x - 4 = 0$, the sum of the roots is $-12$ and the product of the roots is $-4$.

Plugging these values into the formula, we get: Sum of the squares of the roots=(−(−12*12))−2(−4)
=(12*12)+8
=144+8
=152

So, the sum of the squares of the roots of the given equation is 152.

So, we have:

$60 + a$, $120 + a$, and $140 + a$

For a three-term geometric sequence, the ratio between consecutive terms is constant.

Thus, we can write:

$\frac{120 + a}{60 + a} = \frac{140 + a}{120 + a}$

Cross multiplying:

$(120 + a)^2 = (60 + a)(140 + a)$

Expanding both sides:

$14400 + 240a + a^2 = 8400 + 200a + 140a + a^2$

$14400 + 240a + a^2 = 8400 + 340a + a^2$

$6000 = 100a$

$a = 60$

Therefore, the common ratio of the resulting sequence is $60$.

Simplify the inequality:

−8≤X+17<−3−5X

Subtract 17 from all parts of the inequality:

−8−17≤X+17−17<−3−5X−17

−25≤X<−20−5X

Now, isolate X on one side of the inequality:

Add 5X to all parts of the inequality:

5X−25≤X+5X<−20−5X+5X

5X−25≤6X<−20

Subtract 5X from all parts of the inequality:

5X−25−5X≤6X−5X<−20−5X

−25≤X<−20

So, the values of X that satisfy the inequality are in the interval [−25,−20)