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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44 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 ...
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2answers
25 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 ...
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1answer
17 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 ...
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0answers
14 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 ...
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3answers
61 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 ...
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3answers
45 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 ...
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4answers
41 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 ...
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1answer
46 views

Two candidates attempt to solve a quadratic equation of the form x² +p x +q = 0 with wrong value. [closed]

Two candidates attempt to solve a quadratic equation of the form x² +p x +q = 0. One starts with a wrong value of p and finds the roots to be 2 and 6. The other starts with a wrong value of q and ...
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3answers
108 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 ...
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0answers
13 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. ...
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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 ...
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0answers
39 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?
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2answers
103 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. ...
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1answer
33 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
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1answer
64 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 ...
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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 ...
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2answers
24 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 ...
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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 $ ...
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1answer
41 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?
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2answers
67 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? ...
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1answer
54 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 ...
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3answers
35 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 ...
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1answer
162 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 ...
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3answers
43 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 ...
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1answer
60 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 ...
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1answer
76 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 ...
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1answer
30 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 ...
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2answers
152 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 ...
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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 ...
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1answer
42 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 ...
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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 ...
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3answers
174 views

What am i doing wrong when solving this differential equation

$$ f(x) = \frac{\frac{d^2}{dx^2}[e^y]}{\frac{d}{dx}[e^y]} $$ Given that $f(x) = cx$ $$ \frac{c}{2}x^2 + k_1 = \ln(e^y y') $$ $$ k_2\int e^{\frac{c}{2}x^2} dx = e^y $$ $$ y = \ln(k_2\int ...
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2answers
45 views

Constructing a parabola from two coordinate values and a peak height.

I am trying to create a parabola function that mimics projectile trajectory. It will input the following: 2 xy coordinates (the starting projectile position and the landing projectile position) ...
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2answers
31 views

Quadratic Equation Roots Prove

I have a question in my textbook from chapter of quadratic equations from exercise of sum of roots and product of roots that; Prove that the equation $$ a x^2 + b x + c = 0, \quad a > 0 $$ has ...
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1answer
61 views

Does $x^2=83\pmod{101}$ have solutions? without calculating them

Does $x^2=83\pmod{101}$ have solutions? without calculating them. I'm not sure how to tackle this without solving, I tried using chinese remainder and quadratic reciprocity.
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27 views

Calculating speed of an object between time intervals using quadratic function

I'm actually getting stuck with a quite difficult part of a 1D math problem using quadratic functions. An object is propelled with an initial speed, has its altitude given after t seconds by the ...
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2answers
83 views

Calculating speed and distance travelled by an object using quadratic function

I'm actually getting stuck with a tricky part of a math problem using a quadratic function. An object propelled with an initial speed, has its altitude given after $t$ seconds by the quadratic ...
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0answers
66 views

Solving quadratic diophantine equations in two variables

I've looked at the recommended questions, but none of them seem to match my question. Consider the equation $2015 = \frac{(x+y)(x+y-1)}{2} - y + 1$. This can trivially be simplified to $4030 = x^2 + ...
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4answers
67 views

If $a$ is an odd integer then $x^2+x-a = 0$ has no integer solutions

I'm suppose to prove by contrapositive that if $a$ is an odd integer then the equation $x^2+x-a=0$ has no integer solution. By contrapositive: If the equation $x^2+x - a = 0$ has an integer ...
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3answers
40 views

solve for x in the following Quadratic equations

I am not able to solve this problem for my son. Is ther any error in the question itself. $\dfrac{1}{(x+1)} + \dfrac{1}{(x+5)}=\dfrac{1}{(x+2)}+\dfrac{1}{(x+4)}$. Thanks in advance..
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3answers
46 views

How to prove the existence of a minimum of a quadratic function of two variables?

I am given function $$ f(x,y)=Ax^2+2Bxy+Cy^2+2Dx+2Ey+F,\quad\text{where }A>0\text{ and }B^2<AC . $$ Prove that a point $(a,b)$ exists which $f$ has a minimum. I figured out that there is ...
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31 views

area of a sprinkeler.

A rectangular lawn has an area of 677 square meters. Surrounding the lawn is a flower border 4 meters wide. The border alone has an area of 548 square meters. A circular sprinkler is installed in the ...
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4answers
46 views

Complete the square in the form $(px+q)^2+r, p > 0$

I'm going over some completing the square questions and I need to express, in the form: $(px+q)^2+r, p > 0$ the quadratic equation is $16x^2-8x+11$ I know how to get it in the form $p(x+q)^2+r$ ...
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0answers
21 views

Numerical method to find coefficient of quadratic function given target skewness

I have two samples, $X$ and $Y$, and for both I calculate the sample skewness Sk$(X)$ and Sk$(Y)$. My objective is to find $d$ such that, given $Z = X + dX^2$, Sk$(Z) = $ Sk$(Y)$. The coefficient of ...
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1answer
40 views

Solving x^4=a mod p, given a is a quadratic residue

Given prime number $p\equiv 1 \pmod 4$. Prove if $a∈F_p^×$ is a quadratic residue then the congruence $$x^4 ≡ a \pmod p$$ has either no solutions or four solutions. Give examples of each case.
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1answer
40 views

The quadratic $x^2-4kx+3k = 0$ has two distinct roots $m$ and $n$, where $m > n$ and $m - n = m^2+n^2$. What is the sum of all possible values of k?

I was trying to solve the following question: The quadratic $x^2-4kx+3k = 0$ has two distinct roots $m$ and $n$, where $m > n$ and $m - n = m^2+n^2$. What is the sum of all possible values of ...
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2answers
34 views

Let $k$ be a real number. Prove that if the equation $|x^{2} - 3x| = x-2+k$ has two distinct roots, then either $-1 < k < 2$ or $k > 3$?

The title is the problem. The condition "has two distinct roots" is ambiguous, but I assume it to be ``having exactly two distinct roots".
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2answers
34 views

Help finding the exact form through quadratic formula

Question: $$\frac{5m}{2}=2+\frac{1}{m}$$ I have attempted the question but my answer is not correct according to the book. $$\frac{5m^2}{2m}-2=0$$ $$5m^2-2=2m$$ $$5m^2-2m-2=0$$ When I placed my ...
3
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2answers
61 views

limits of $x$ for $( x-a)(x-b)>c$

After solving a quadratic inequality, I've ended up getting the solution in the format- $(x-a)(x-b)>c$, where $a,b,c$ are real constants, then how can I decide on the limits of $x$. will the ...
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0answers
41 views

How to show that $2^x$ is not in $O(x^2)$?

This is from Discrete Mathematics and its Applications I am working on 2e. I knew right off the bat from previous computer science courses that 2^x is not in O(x^2). I am having a difficult time ...