Raven Gardner

2022-04-06

Find the sum of all positive integers k for which $5{x}^{2}-2kx+1<0$ has exactly one integral solution.

My attempt is as follows:

$(x-d\frac{2k-\sqrt{4{k}^{2}-20}}{10})(x-d\frac{2k+\sqrt{4{k}^{2}-20}}{10})<0$

$(x-d\frac{k-\sqrt{{k}^{2}-5}}{5})(x-d\frac{k+\sqrt{{k}^{2}-5}}{5})<0$

$x\in (d\frac{k-\sqrt{{k}^{2}-5}}{5},d\frac{k+\sqrt{{k}^{2}-5}}{5})$

As it is given that it has got only one integral solution, so there must be exactly one integer between$d\frac{k-\sqrt{{k}^{2}-5}}{5}$ and $d\frac{k+\sqrt{{k}^{2}-5}}{5}$ .

Let$x}_{1}=d\frac{k-\sqrt{{k}^{2}-5}}{5$ and $x}_{2}=d\frac{k+\sqrt{{k}^{2}-5}}{5$ , then $\left[{x}_{2}\right]-\left[{x}_{1}\right]=1$ where [] is a greater integer function.

But from here, how to proceed? Please help me in this.

My attempt is as follows:

As it is given that it has got only one integral solution, so there must be exactly one integer between

Let

But from here, how to proceed? Please help me in this.

Frain4i62

Beginner2022-04-07Added 16 answers

Your idea is good; you want to find all positive integers k for which there is precisely on integer between the roots of

$5{x}^{2}-2kx+1=0.$

Then the distance between the roots can be at most 2, where the distance between the roots is precisely

$\frac{1}{5}\sqrt{{(-2k)}^{2}-4\cdot 1\cdot 5}=\frac{25}{\sqrt{{k}^{2}-5}},$

as you already found. This is at most 2 if and only if$\sqrt{{k}^{2}-5}\le 5$ , or equivalently $k\le 5$ . This leaves only 5 values of k to check.

Then the distance between the roots can be at most 2, where the distance between the roots is precisely

as you already found. This is at most 2 if and only if

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) $f(x)=-3x+4$

b) $f\left(x\right)=-3{x}^{2}+7$

c) $f(x)=\frac{x+1}{x+2}$

?

$d)f\left(x\right)={x}^{5}+1$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.$g(x)=5-2x$

c.$h\left(x\right)=\frac{2}{x}+3$

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 theseWhat 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)3How to graph $y=\frac{1}{2}x-1$?

How to graph $y=2x+1$ using a table?

simplify $\sqrt{257}$

How to find the vertex of the parabola by completing the square ${x}^{2}-6x+8=y$?

There are 60 minutes in an hour. How many minutes are there in a day (24 hours)?

Write 18 thousand in scientific notation.