Questions on quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

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0
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4answers
48 views

For what $k$ is $f(x) = kx^2-2x+k$ negative for all values of $x$?

What are the values of $k$ for which the quadratic function $f(x) = kx^2-2x+k$ is negative for all values of $x$? The values of $k$ should definitely be negative.
1
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3answers
39 views

If $a>0, a \neq 1,\,$ then the equation $\,2 \log_x (a)+ \log_{ax} (a) + 3\log_{a2x}, (a)=0,\,$ has how many real roots?

If $\,2 \log_x (a)+ \log_{ax} (a) + 3\log_{a2x}, (a)=0,\,$ then the equation has how many real roots? The above problem is a quadratic equation problem.
0
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3answers
46 views

quadratic function positive

Put constraints on a quadratic function. I know that for $x > 0$ then $ax^2 + bx + c > 0$ I read around but I just found positive for all $x$. Thanks a lot
1
vote
1answer
23 views

If $\alpha_i$ are the roots of $x^n + nax−b = 0$ then show that $\prod_{1< i \le n} (\alpha_1 -\alpha_i)=n(\alpha_1^{n-1}+a)$

If $\alpha_i$ are the roots of $x^n + nax−b = 0$ then I would like to show that $$\prod_{1< i \le n} (\alpha_1 -\alpha_i)=n(\alpha_1^{n-1}+a).$$ The only thing I could think is differentiating $x^...
0
votes
2answers
57 views

The expression $(x+a)(x+1991) +1$ can be factored as a product $(x+b)(x+c)$ where $b,c$ are integers

I need to solve following problem, but don't know to start with, please provide hints and solutions. Find all integer values of $a$ such that the quadratic expression $(x+a)(x+1991) +1$ can be ...
2
votes
2answers
30 views

What is the coordinate of the maximum value of a quadratic function given by two points and axis?

There are only three pieces of information available: the graph passes through (0,0) and (6,0) the symmetry axis is $x$ = 3 the graph is downward My attempt: I've tried to work on ...
0
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2answers
19 views

Find $m$ so that given equation has real roots

We are given the following equation: $$x^4 - (2m - 1)x^2 + 4m - 5 = 0, m \in \mathbb{R}$$ Find all $m$'s so that the given equation has real roots. I thought I only had to put $\Delta \geq 0 $. ...
0
votes
2answers
20 views

Integral values of $m$ for rational roots.

Consider the function $f(x)=mx^2+(2m-1)x+(m-2)$. Choose the correct options: $(A).$ If $f(x)>0$ for all $x \in R$, then $m \in (- \infty, -1/4)$ $(B).$ The number of integral values of $m$ ...
0
votes
5answers
53 views

Formula for factorization of a Quadratic Equation?

To be clear I am looking for an equation to go from $$Ax^2 + Bx + C = 0$$ To $$(Dx + E)(Fx + G) = 0$$ And I need it to be able to be done in a computer as it will be going in my app. Thanks in ...
0
votes
1answer
34 views

Express $c$ and $d$ in terms of $m$ where $c$ and $d$ are zeroes of $f$ where $m > -2$

Let $$f(x) = x^2 - mx -(6m^2+25m+25)$$ where $m > - 2$ It can be shown that $f(x)$ has two zeroes. Suppose we have $c,d \in \mathbb R$ s.t. $c < d$ and $f(c) = f(d) = 0$, express $c$ and $d$ ...
1
vote
0answers
18 views

Derivative of quadratic form involving singularity

This might be a silly question, but i have been really curious about the following: Consider the following function seen thru a single variable, say $\alpha$: \begin{equation} f(\alpha) = \mathbf{x}^...
0
votes
1answer
24 views

How to solve $7.51\tan{\theta} - 2.656(\sec{\theta})^2=0$

I'm trying to solve $7.51\tan{\theta} - 2.656(\sec{\theta})^2=0$ and the way that it's been done in my notes is by somehow changing the equation to $7.51\tan{\theta} - 2.656(\tan{\theta})^2 - 2.656=0$ ...
0
votes
2answers
22 views

For what values of k the expression will be a perfect square?

the question is the expression $kx^2 +(k+1)x +2$ will be a perfect square of a linear polynomial for what values of k . I am unable to understand the concept used in this question for finding the ...
3
votes
4answers
78 views

Solving Quadratic system of equations

Solve this system of equations: $$(1) \quad 0=-10x^2-9xy+50x-25y-7y^2+5$$ $$(2) \quad 0=-5x^2-17xy+25x+50y-14y^2+7$$ Shame on me but I'm failing to solve this system. I can't see a short (...
1
vote
2answers
38 views

Find the stopping distance

(original question, see edits below for full context) After much frustration, I have figured out a function which maps velocity during acceleration/deceleration for my project. $$\text{velocity} =s+\...
1
vote
1answer
24 views

How to solve this system of non-linear equations of second order?

I have a system of three equations: $$a_1- (b_1x+cx^2-cx) + (dx - x^2 + x) - yz = 0 $$ $$a_2- (b_2x+cx^2-cx) + (dx - x^2 + x) - (y+1)z = 0 $$ $$a_3- (b_3x+cx^2-cx) + (dx - x^2 + x) - (y+2)z = 0 $$ ...
6
votes
5answers
836 views

Approximation to unsolvable system of equations

I am working on a project and need to find the "closest" numerical values that satisfy the following equations: \begin{equation} \left\{ \begin{array}{} A \cdot C = \frac{1}{2} \\ A ...
190
votes
21answers
21k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
0
votes
2answers
34 views

To find the range of x. [closed]

Let $a,b,c,d$ be four distinct real numbers such that they are in AP.If $$2(a-b)x(b-c)^2+ (c-a)^3= 2(a-d)+ (b-d)^2 + (c-d)^3$$ then find the range of $x$.
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0answers
51 views

To prove that roots of a polynomial are real. [closed]

Given three polynomials $$ f(x)=x^2+a_1x+b_1, \\ g(x)=x^2+a_2x+b_2, \\ h(x)=x^2+a_3x+b_3 $$ such that $$a_1a_2a_3=b_1b_2b_3>1.$$ Prove that one of these polynomials has 2 distinct real roots.
11
votes
7answers
745 views

Factoring polynomials with a 2nd degree coefficient greater than $1$

I'm learning how to factor polynomials, but I'm having a hard time understanding the approach when the 2nd degree coefficient is greater than $1$. For example, when I begin to factor $12k^4 + 22k^3 - ...
0
votes
0answers
19 views

How to shorten dot product

I would like to shorten a dot/scalar product: $$f(s)=sP_1+s^2P_2+\big((P_2-P_1)^TsN_1\big)N_1$$ Here $s$ are scalars, $P$ are points and $N$ are unit normal vectors in $R^3$. The function $f(s)$ ...
2
votes
2answers
48 views

Solving equations system: $xy+yz=a^2,xz+xy=b^2,yz+zx=c^2$

Solve the following system of equations for $x,y,z$ as $a,b,c\in\Bbb{R}$ \begin{align*}xy+yz&=a^2\tag{1}\\xz+xy&=b^2\tag{2}\\yz+zx&=c^2\tag{3}\end{align*} My try: Assume that $x,y,z\...
0
votes
1answer
49 views

When are we permitted to multiply or divide both sides of an equation by a constant?

For example, let's consider the quadratic equation $-3x^2 + 6x -2 = 0$. Multiplying both sides by $-1$, we get the equation $3x^2 - 6x +2 = 0$. The graph of the above equations are different even ...
3
votes
1answer
26 views

Solving the quadratic formula to determine stability of a system

I am trying to solve the $2\times 2$ matrix $$\begin{bmatrix} 0 &1 \\ -k &-b \end{bmatrix}$$ for a relationship between the variables $k$ and $b$ to determine when a system is stable. ...
1
vote
1answer
75 views

Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution

Find the values of 'b' for which the equation $$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution. =$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$ My try: After removing the ...
1
vote
2answers
51 views

Find the value of $P(1)$

Let $P (x) = x^2 + bx + c$, where $b$ and $c$ are integer. If $P(x)$ is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$, find the value of $P(1)$. I am not being able to solve ...
2
votes
2answers
71 views

What are the type of roots for the equation?

The equation is $7x²-(7\pi+22)x+22\pi=0$. I am not sure if i should put the value of $\pi=$ (22/7)or (3.14) or even if I should do that.. The question asks for the type of roots rational or ...
0
votes
1answer
44 views

Quadratic number pattern equation

May I know how do I form a quadratic number pattern equation? I cant seem to form one on my own. 1500, 1519,1536, 1551,1564.
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2answers
36 views

Solving an equation involving the root of a floor function of x

$x- ⌊\sqrt{x} ⌋^2 =10$. Prove that $x=35$ where $⌊x⌋$= floor of x
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1answer
34 views

Quadratic Equation Based Problem:Prove either $a = 2l$ & $b = m$ or $b + m = al$

If by eleminating $x$ between the equation $x² + ax + b = 0$ & $xy + l (x + y) + m = 0$, a quadratic in $y$ is formed whose roots are the same as those of the original quadratic in $x$. Then ...
0
votes
1answer
480 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
6
votes
0answers
91 views

Is a finite number of quadratic equations in two variables sufficient to solve for the two variables?

Let's say I’m trying to solve a Diophantine problem in two positive integers, $y$ and $q$. Furthermore, let’s say I can derive an extremely large (read: arbitrary) number of equations of the form $$ay^...
2
votes
3answers
83 views

Find $x_1^n+x_2^n$ on any quadratic equation, general case.

I have a simple quadratic (with $x^2$) equation, x can Be complex too: $$x^2+x+1=0$$ But it could be any equation, the equation above is just an example. I need to compute $x_1^{10}+x_2^{10}$, but ...
6
votes
4answers
623 views

'Concentric' parabolas — two parabolas that have a constant vector distance

I am trying to 'draw' a two-dimensional path in the shape of a semi circle with thickness d in the xy-plane. The way I would like to do this is to have two parabolas, $f(x) = ax^2$ and $g(x)= bx^2 +cx ...
0
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0answers
27 views

How to rewrite equation to get a quadratic patch

I would like to understand the given rewrite or transform from one equation to another. This is the original equation: $$p^*(q)=(u,v,w)\left( \matrix{q-n_i\big((q-x_i) \cdot n_i \big) \cr q-n_j \big((...
4
votes
4answers
120 views

$f(x)$ is a quadratic polynomial with $f(0)\neq 0$ and $f(f(x)+x)=f(x)(x^2+4x-7)$

$f(x)$ is a quadratic polynomial with $f(0) \neq 0$ and $$f(f(x)+x)=f(x)(x^2+4x-7)$$ It is given that the remainder when $f(x)$ is divided by $(x-1)$ is $3$. Find the remainder when $f(x)$ ...
1
vote
0answers
39 views

Trigonometric roots of a cubic

Let the product of the sines of the angles of the triangle is $\frac{2}{3}$ and the product of their cosines is $\frac{1}{9}.$ If $\tan A$ , $\tan B$ and $\tan C$ are the roots of the cubic, find the ...
0
votes
0answers
29 views

Quadratic Equation w/ 3 unknowns

A quadratic equation -> x^2+2hx+h^2-k^2=0 is given and the question is to find the roots. I'm a noob so don't expect much. The 2nd question is -> If the sum and the difference of the roots of the ...
1
vote
4answers
43 views

where is my mistake with this equation ot hyperbolic problem?

Let parabola $\Gamma_{1}:$$y^2=4x$,and hyperbolic curve $\Gamma_{2}$: $x^2-y^2=1$.it is well known this two is symmetric.so the two point $A$ and $B$ about $x$ axial symmetry,or mean $x_{A}=x_{B}$....
3
votes
2answers
43 views

Solution of equation

If $f(x) = x^2 - 2ax + a(a+1)$ , $f:[ a, \infty] \to [a,\infty]$ . If one of the solution of the equation $f(x)=f^{-1}(x)$ is $5049$ , then what may be the other solution ? My WORK: I found the ...
2
votes
1answer
30 views

Find $ m \in \mathbb{Z} $ for which $ x_1 $ and $ x_2 $ are integers

$$ (m+1)x^2 - (2m+1)x - 2m = 0 $$ $$ m \in \mathbb{R}-\{-1\} $$ Find $ m \in \mathbb{Z} $ for which $ x_1 $ and $ x_2 $ (the solutions of equation, the roots) are integers ($x_1,x_2 \in \mathbb{Z}$) ...
1
vote
1answer
62 views

show the quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$ is quasi-concave

I have a quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$, with $x_i$ nonnegative and $A \in[0,1)$. And w.l.o.g. we can normalize $x_i's$ to between 0 and 1. In ...
-1
votes
1answer
30 views

If $w$ is an imaginary cube root of unity, then the polynomial whose roots are $2w+3w^2$ and $2w^2 + 3w$ is?

What polynomial with complex coefficients has the following as its roots? $2w+3w^2$ and $2w^2 + 3w$ I have tried doing this all the ways I know of, still can't get my pen over it... Can you guys ...
1
vote
3answers
64 views

Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$

Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$ Solutions are $x=2$ or $x=-1$. But $x=2$ does not satisfy $\sqrt{x+2}+x=0$.Because $\sqrt{4}+2 \neq0$ So does it mean that they are ...
2
votes
3answers
49 views

Quadratic Functional equations.

Suppose $f$ is a quadratic ploynomial, with leading cofficient $1$, such that $$f(f(x) +x) = f(x)(x^2+786x+439)$$ For all real number $x$. What is the value of $f(3)$?
0
votes
1answer
20 views

Can the two unknowns n and d (in the equations below) be found by using the simultaneous equations method

The two unknowns are n=3 and d=30. So the answers can be easily found by trial and error. Is it possible to find d and n using the simultaneous equation method ? If so could someone do the calculation....
1
vote
1answer
23 views

Using discriminant to prove that there are real solutions to a quadratic

Question: Prove that there are two solutions of the equation $x^2 + kx = 3 - k $ The obvious solution would be finding the value of $b^2 - 4ac$ $x^2 + kx + k - 3=0$ $b^2 - 4 ac$ =$k^2 - 4(1)(...