A high school quadratic problem"If a1,a2,a3,…, an are distinct non-zero numbers such that(∑k=1n−1{{ak}2})x2+2(∑k=1n−1akak+1)x+(∑k=2n{{ak}2})≤0

From Cauchy-Schwarz inequality,(2∑limitsk=1n−1akak+1)2−4∑limitsk=1n−1ak2∑limitsk=2nak2≤0 so:(∑limitsk=1n−1ak2)x2+(2∑limitsk=1n−1akak+1)x+∑limitsk=2nak2=0 has only one solution. We can get (2∑limitsk=1n−1akak+1)2−4∑limitsk=1n−1ak2∑limitsk=2nak2=0, then ak+1ak=c, here c is a constant.

Answered: "A high school quadratic problem"If a1,a2,a3,…, an are..."

High School
Algebra
Algebra I
Quadratic function and equation

Gary Johnson

Answered question

2022-03-04

A high school quadratic problem
"If

a_{1}, a_{2}, a_{3}, \dots

, an are distinct non-zero numbers such that

(\sum_{k = 1}^{n - 1} {{a_{k}}^{2}}) x^{2} + 2 (\sum_{k = 1}^{n - 1} a_{k} a_{k + 1}) x + (\sum_{k = 2}^{n} {{a_{k}}^{2}}) \leq 0

Answer & Explanation

Yosef Krause

Beginner2022-03-05Added 8 answers

From Cauchy-Schwarz inequality,

{(2 \sum lim i t s_{k = 1}^{n - 1} a_{k} a_{k + 1})}^{2} - 4 \sum lim i t s_{k = 1}^{n - 1} a_{k}^{2} \sum lim i t s_{k = 2}^{n} a_{k}^{2} \leq 0

so:

(\sum lim i t s_{k = 1}^{n - 1} a_{k}^{2}) x^{2} + (2 \sum lim i t s_{k = 1}^{n - 1} a_{k} a_{k + 1}) x + \sum lim i t s_{k = 2}^{n} a_{k}^{2} = 0

has only one solution. We can get

{(2 \sum lim i t s_{k = 1}^{n - 1} a_{k} a_{k + 1})}^{2} - 4 \sum lim i t s_{k = 1}^{n - 1} a_{k}^{2} \sum lim i t s_{k = 2}^{n} a_{k}^{2} = 0

, then

\frac{a_{k + 1}}{a_{k}} = c

, here c is a constant.

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Algebra I

Find the volume V of the described solid S
A cap of a sphere with radius r and height h.
V=??
Whether each of these functions is a bijection from R to R.
a) $f (x) = - 3 x + 4$
b) $f (x) = - 3 x^{2} + 7$
c) $f (x) = \frac{x + 1}{x + 2}$
$d) f (x) = x^{5} + 1$
?
In how many different orders can five runners finish a race if no ties are allowed???
State which of the following are linear functions?
a. $f (x) = 3$
b. $g (x) = 5 - 2 x$
c. $h (x) = \frac{2}{x} + 3$
d. $t (x) = 5 (x - 2)$
Three ounces of cinnamon costs $2.40. If there are 16 ounces in 1 pound, how much does cinnamon cost per pound?
A square is also a
A)Rhombus;
B)Parallelogram;
C)Kite;
D)none of these
What is the order of the numbers from least to greatest.
$A = 1.5 \times 10^{3}$ ,
$B = 1.4 \times 10^{- 1}$ ,
$C = 2 \times 10^{3}$ ,
$D = 1.4 \times 10^{- 2}$
Write the numerical value of $1.75 \times 10^{- 3}$
Solve for y. 2y - 3 = 9
A)5;
B)4;
C)6;
D)3
How to graph $y = \frac{1}{2} x - 1$ ?
How to graph $y = 2 x + 1$ using a table?
simplify $\sqrt{257}$
How to find the vertex of the parabola by completing the square $x^{2} - 6 x + 8 = y$ ?
There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?
Write 18 thousand in scientific notation.

Didn't find what you were looking for?