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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2
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2answers
59 views

Help me to prove this statement about quadratic equations? (from Gelfand's Algebra).

$ x^2+px+q=0 ${p,q are integers; a,b are roots}. Prove $a^n+b^n$(n is any natural number) is an integer. This is the third part of the problem.I have previously proved that $a^2+b^2$ and $a^3+b^3$ ...
0
votes
2answers
67 views

Difference between two real roots with uniformly distributed coefficents

I have a question that first I need to know what is happening, but then I also need to code it in a program called APPL, which is an extension from Maple18 that I really have never used, yet I have ...
0
votes
2answers
71 views

calculate the intersection of two number series

I have a series of numbers. It is in the form of a parabola. This series is guaranteed to have at least one perfect square within it (edited I thought there was only one). The second series is also a ...
1
vote
3answers
34 views

Relationship between roots and equations

I'm stuck on topic of relationship between roots and equations. The roots of $x^2 -2x +3 =0$, are $\alpha$ and $\beta$. Find the equation whose roots are : 1- $\alpha+2$, $\beta+2$ 2- $\alpha^2$, ...
7
votes
4answers
310 views

Why are there four solutions to $x^2-2x-8=0$ in $\mathbb{R}$? Or am I wrong?

It might be a very trivial question to ask but why do we get four different solutions for a quadratic equation using these two methods? $x^2-2x-8=0$ We see that factors are $(x-4)$ and $(x+2)$ so ...
2
votes
2answers
51 views

How do I complete the square when the $x^2$ has a coefficient greater than $1$?

For homework we are doing completing the square and a few of them have coefficients greater than one. For example one of the quadratic equations we have to complete the square of is $-2x^2-7x-2$. All ...
0
votes
1answer
33 views

Complex Roots Of a equation - Equilateral triangle

$z_1$ and $z_2$ are the roots of $3z^2+3z+b=0$.If $O(0),A(z_1),B(z_2)$ is an equilateral triangle then what will be the value of b ? My approach:I took $z_1=m_1+in_1$ and $z_2=m_2+in_2$ and proceeded ...
0
votes
0answers
23 views

Alternative method for y-vertex calculation

So, I've been wondering the following: If you can determine the x coordinate of the vertex of a quadratic function by averaging the x coordinates of both roots, would it be possible to determine the ...
0
votes
1answer
48 views

Derive Equation from the set of values

I have a set of values $$\begin{array}{|c|c|} \hline\text{$X$} & \text{$Y$} \\ \hline 1 & 2 \\ \hline 2 & 10 \\ \hline 3 & 30 \\ \hline 4 & 68 \\ \hline \end{array}$$ ...
1
vote
1answer
289 views

Show that quadratic is positive for all real values of x

I have been asked this question: Show that $x^2 + 2px + 2p^2$ is positive for all real values of $x$. I've worked it out like so: Discriminant = $(2p)^2 - (4\times 1\times(2p^2)) = 4p^2 - 8p^2$ I ...
1
vote
2answers
34 views

Figuring domain of constant $a$ in a equation with some condition

Here is what questions says Question: If $a\in \mathrm{R}$ and the equation $-3(x-[x])^2+2(x-[x])+a^2=0$ (where $[\cdot]$ denotes the greatest integer $\leq x$) has no integral solutions, then all ...
1
vote
1answer
41 views

Quadratic Functions Word Problem Help

Two numbers differ by 18. determine the two numbers if the sum of their square is 3860.
0
votes
1answer
26 views

Why is this answer wrong? (point of intersection between parabola and line)

Question: Use the discriminant to determine the number of points of intersection of the line $y=3x+5$ and the quadratic functions $f(x)=3x^2-2x-4$. Solve to find the points of intersection. ...
-1
votes
3answers
37 views

Find the speed of a jet given the time of travel back and forth

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
0
votes
2answers
42 views

Converting from factored to standard form: why is this answer wrong?

Converting the equation $$y=-2(x-2+\sqrt{5})(x-2-\sqrt{5})$$ to standard form seems to give $$-2x^{2\space }+3.528x+6.4171392.$$ My handout tells me that the answer is different. What is wrong here? ...
2
votes
1answer
158 views

Why is this answer wrong? (quadratic functions)

Question: Determine the quadratic function that has the given roots (x-intercepts) and passes though the given point. $x=2+\sqrt{5}, x=2 - \sqrt{5}$. The given point is $(2,10)$. This is my ...
0
votes
0answers
30 views

How do I solve this quadratics problem? [duplicate]

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
2
votes
4answers
67 views

How to factor quadratics $(x^2 + 4x + (-357) = 0)$

I need to find $2$ factors of $-357$, which add up to $4$. Obviously one number is positive and the other is negative. I understand this and I know the factors can be $21$ and $-17$; but, how do I ...
0
votes
1answer
49 views

Which are the conditions for a biquadratic equation to have 4 different roots?

Which are the conditions for a biquadratic equation to have 4 different roots in R? I think D>0, If we have $$t=x^2$$ then t>0. Is there any other condition?
1
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0answers
78 views

Minimum Curvature Path

Let's say we are given a closed race track with a given and constant width. I am to implement an algorithm which finds both shortest path trajectory and minimum curvature trajectory for the car. I ...
0
votes
1answer
61 views

Find Minimizer and Minimum Value for a Function

I am trying to work through some problems to find the minimizer and minimum value of a function. The book I am using doesn't have a clear cut example and I can't seem to find a good example online ...
1
vote
2answers
28 views

fitting a quadratic to 3 coordinates

say I have 3 points on the plane (Cartesian coordinate system), (a,b), (c,d) and (e,f), I am fairly certain that there is one unique quadratic curve which passes through each point. what is the ...
1
vote
1answer
18 views

Finding the positive component of a constant in a quadratic equation.

Can you help me to do this question:it is from a past cambridge exam paper Find the positive constants $a$ and $b$ such that $x^4+9/x^4 =[x^2-a/x^2 ]^2+b$ for all non-zero values of $x$. Hence write ...
0
votes
0answers
16 views

Determine the multi-dimensional relationship given the data

I have a dependent variable - A and 3 independent variables, H,V and N I have a data for all the variables and dependency relationship is based on my operational knowledge. I'd like to know what ...
0
votes
3answers
64 views

How to solve this quadratic equation?

So I've got this quadratic equation and am totally unable to solve it. Can someone tell me how to do it? $$\frac{a}{ax-1} + \frac{b}{bx-1} = a + b,$$ where $x$ is not equal to $\frac{1}{a}$ or ...
1
vote
3answers
57 views

Quadratic equation involving floor function.

If equations $x^2-3x+4=0$ and $ 4x^2-2\lfloor3a+b\rfloor x+b=0\space (a,b\space\epsilon\space R) $ have a common root then the complete set of values of $a$ is ? I have not yet been able to develop ...
0
votes
4answers
49 views

If $a,b$ are the roots of the equation $2 x^2 -3 x +1 = 0$, find an equation whose roots are $a/(2b +3)$, $b/(2a +3)$

If $a,b$ are the roots of the equation $2 x^2 -3 x +1 = 0$, find an equation whose roots are $a/(2b +3)$, $b/(2a +3)$ I was practicing quadratic equation questions online but I am stuck on this ...
0
votes
3answers
115 views

continued fraction of the roots of $x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$

I would like to find the continued fraction expansion of the roots of: $$x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$$ Eq 1.6 from [1] What makes this problem so ...
0
votes
0answers
17 views

Constrained Motion Study

I'm working on a motion study for a disk moving within a mechanical enclosure and I'm having trouble reducing my equations. The system can be defined as 4 circles which are bound inside each other. ...
0
votes
1answer
33 views

Condition for roots of the equation to be real.

Show that for $ 3 > y_1 >0 $ the roots of the equation $$(y_1-2)x^2-(8-2y_1)x-(8-3y_1)=0$$ are real, where $y_1$ is a constant. Due to my difficulties in doing this I would be grateful for ...
0
votes
0answers
40 views

Proof of the Quadratic formula [duplicate]

Prove: $ax^2+bx+c=0 \implies x=(−b±\sqrt{b^2−4ac})/2a$ I know it's probably simple just can't get my head around it?
1
vote
2answers
107 views

Solve for “x” and “y” [duplicate]

What would be the easiest way to solve the following set of equations:$$ x + y^2 = 7 $$$$ x^2 + y = 11$$ I've been trying substitution method but end up in a $4$th degree bi-quadratic equation. ...
1
vote
1answer
63 views

Quadratic equation!Given that 1/3 is one of the root

$$px^2-4x+p-2=0$$ root=1/3 can anyone tell me how to do step by step im stuck in the middle $$1/p+p-10/3=0 $$ :D
0
votes
1answer
72 views

How can I convert a quadratic equation to a normalized quadratic equation?

I have a quadratic equation (for a power curve) which produces a value for efficiency at a fixed number of Watts: (1) $e = E_0 + E_1 x + E_0 x^2$ I have a second quadratic equation which should work ...
0
votes
1answer
22 views

Matrix product equals O

I'm stuck in this question : \begin{bmatrix} x & 4 & -1 \end{bmatrix} \begin{bmatrix} 2 & 1 & 0\\ 1 & 0 & 2\\ 0 & 2 & 4 \end{bmatrix} \begin{bmatrix} x\\ 4\\ 1 ...
0
votes
2answers
30 views

Quadratics: Word Problem (Height, Width)

We're learning about Quadratics, but I'm not exactly sure how this applies to it: $\dfrac{w + h}{w} = \dfrac{w}{h}$. If the height is 16 inches, what is its width? (Round to the nearest tenth.) Can ...
0
votes
1answer
31 views

solving the equation $x^{n}-dy^{n}=1 $ in integers

how could we solve the equation $x^{n}-dy^{n}=1 $ by knowing the continued fraction expansion of $ d^{1/n} $ ?? in case $ n=2 $ is pell's equation if I divide all by $ y^{n} $ then $ ...
0
votes
1answer
61 views

track and field word problem

Word Problem: A track and field runner saves 1 hour by covering 112 km at a rate which is 2 kmph greater than the usual rate. How many hours does he usually take to travel this distance?
3
votes
2answers
68 views

Find $m,\alpha>0,\alpha\in\mathbb Z[\sqrt{m}]$ s.t. $\forall p,q,r,n\in\mathbb Z[\sqrt{m}]$, $p^2+q^2+r^2\neq\alpha n^2$

Let us assume that $\alpha, p,q,r,n\in\mathbb Z[\sqrt{m}]$ and $m\in\mathbb{N}$ is a square free integer. is it possible to choose $\alpha,m$ such that the following equation is $never$ satisfied? ...
0
votes
1answer
57 views

Integer solution of second degree equation

We all know that on $\mathbb{R}$ the solution of a second degree equation in the form $Ax^2 + Bx + C$ is given by: \begin{equation} \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \end{equation} Now, be $A, B, C ...
3
votes
3answers
37 views

quadratic formula when a and c are positive?

How can you do the quadratic forumla when a and c are positive? I am in a calc class trying to find when the velocity is 0 with a given quadratic equation. But when a and c are positive you get a ...
1
vote
1answer
425 views

Find range of values for 'k' that gives this equation 2 distinct real roots?

I stuck on a question which is asking me to find the range of values for k. the question is : By considering the discriminant, or otherwise, find the range of values of 'k' that gives the equation 2 ...
1
vote
3answers
45 views

Quadratic formula question: Missing multiplying factor of A?

I have a very simple problem which must have a simple answer and I was wondering if anyone can point out my error. I have the following quadratic equation to factor: $2x^2+5x+1$ Which is of the ...
1
vote
1answer
69 views

How to solve inequality problem without factoring or quadratic equation

I'm tutoring someone, and I'm stuck on one of her problems. The equation is $\sqrt{x+14}\le x-16$. She hasn't been taught the quadratic formula or how to factor these problems yet. Is there a way to ...
3
votes
1answer
93 views

Quadratic surfaces: Coordinates and radius( Non origin)

So I have a problem figuring out how to find the coordinates and radius to quadratic equations that are not in the form of $$(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 $$ Where the coordinates are going ...
0
votes
1answer
32 views

Prove that quadratic equation is 0 for any integer $a, b, c$

Prove that there exists a number $r$ such that $ar^2 + br + c = 0$ for any given integers $a, b, c$. I'm stuck on this. Particularly, I see it problematic as $r$ can probably be an irrational or ...
0
votes
2answers
409 views

what numbers multiply to 1 but add to negative 4

I have math hw on writing quadratic equations. You have to write them based on the parabola given in vertex form standard form and intercept/factored form. For the intercept form one step is to find a ...
0
votes
0answers
21 views

Quadratic Polynomial Formula [$ax^2+b_1x+c$ vs. $ax^2+2b_2x+c$]

I thought the quadratic formula was $ax^2+b_1x+c$. However, in my linear algebra book when they deal with $x^TKx$, they use the formula $ax^2+2b_2x+c$. I understand that $b_1$ = $2b_2$, but what is ...
-2
votes
1answer
48 views

Constructing a quadratic equation [closed]

So basically I have been assigned a question that involves constructing a quadratic equation from scratch and graphing it. So here are the details. We are designing a water arc fountain, and it has a ...
0
votes
2answers
51 views

Explanation of double answer

A man is employed to count the total sum of $10710$ . He counts at the rate of $180$ per minute for half an hour. After that he counts at the rate of $3$ less every minute than the preceding ...