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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1answer
29 views

Equation of a curved line from a graph

I am trying to calculate an equation to represent the graph attached to this question. It's an extract from a take-off performance graph used in aviation. The second graph shows how it is used. The ...
1
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1answer
37 views

Formulate quadratic equation

Here is an equation $$r^3-5r^2+8r-4=0$$ Is there a way I can formulate a quadratic equation from this? Sorry if the question seem dumb, I am stuck and I can't figure a way out.
4
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4answers
77 views

Real roots of the equation $\frac{18}{x^4} + \frac{1}{x^2} = 4$

I'm struggling a bit on the best method to find the real roots of the above equation. I ended up obtaining an equation of: $4x^4 - x^2 - 18 = 0$. Is this correct? From there on, how should I ...
0
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1answer
9 views

Solution space for quadratic equations with nilpotent matrices

Let ${\bf w}\in\mathbb{R}^3$ and ${\bf N}\in\mathbb{R}^{3\times 3}$ be a nilpotent matrix with degree 3. Consider the following system of quadratic equations, $$ \begin{align} {\bf w}^\top{\bf w} ...
0
votes
1answer
17 views

Maximize two variables function subject to quadratic constraint

Two mariners end up on a island, with 1800 pounds of food to share, i.e. $F1 + F2 = 1800$. I'm expected to maximize the social welfare function given by $W=U1^{0.25}*U2^{0.75}$ where $U1=\sqrt{F1}$ ...
0
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0answers
11 views

Solution to system of multivariate quadratic equations

I aim to find a vector ${\bf w}\in\mathbb{R}^N$ such that it solves the following system of quadratic equations, $$\forall i,j\quad {\bf w}^\top {\bf A}^{ij}{\bf w} = B_{ij}$$ where ${\bf ...
-1
votes
4answers
29 views

Projectile motion quadratic relations [on hold]

An arrow is show upwards from ground level. After one second its height is $55$ m. and after $2$ seconds, its height is $100$ m. determine the maximum height and the time it takes to hit ...
0
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2answers
31 views

Q: Quadratic Division - How to divide two quadratics?

My studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula However while I have been successful in ...
1
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4answers
53 views

finding real roots by way of complex

I was given $$x^4 + 1$$ and was told to find its real factors. I found the $((x^2 + i)((x^2 - i))$ complex factors but am lost as to how the problem should be approached. My teacher first found 4 ...
1
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2answers
27 views

Why coefficients have to be proportional for two quadratic functions to have the same roots?

We have the next two quadratic functions: $ ax^2 + bx + c = 0 $ $ mx^2 + nx + p = 0 $ If $ a/m = b/n = c/p $ then they have the same roots. What is the intuition behind this statement?
1
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1answer
29 views

How to expand $x_1^3 + x_2^3$ with the parameters of quadratic equation

Given: $X_1$ and $X_2$ are the roots of the equation $ax^2+bx+c = 0$ $a\neq 0$ expand $X_1^3 + X_2^3$ using the parameters a,b and c Here's what I tried to do: $X_1^3 + X_2^3 = $ $(X_1\cdot ...
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3answers
56 views

proving an inequality related to $AM\ge GM$

$$a^2+ab+b^2\ge 3(a+b-1)$$ $a,b$ are real numbers using $AM\ge GM$ I proved that $$a^2+b^2+ab\ge 3ab$$ $$(a^2+b^2+ab)/3\ge 3ab$$ how do I prove that $$3ab\ge 3(a+b-1)$$ if I'm able to prove the ...
1
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2answers
27 views

Condition for roots to lie in certain intervals

The set of values of $p$ such that both the roots of the equation $$f(x)=(p−5)x^2−2px+(p−4)=0$$ are positive and one of the roots is less than $2$ and the other root lies between $2$ & $3$ ...
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2answers
88 views

$p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$; $\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}} = ? $

Let $p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$ The sum $$\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}}$$ Can be expressed as $\frac{a}{b}$. Where $\lfloor \cdot \rceil$ denotes the ...
0
votes
2answers
30 views

How to solve a quadratic inequality that acts like a quadratic equality?

This will be largely a trivial question. But how do I solve an inequality like this: $3x^4 - 4x^2 + 1>0$ ? Of course, I can treat it like a quadratic inequality by saying $t=x^2$ So I can solve ...
0
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1answer
25 views

If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then the value of m is equal to to?

If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then the value of m is equal to to ? I drew the graph of $|x^2+4x+3|$.I found that that for the given condition $mx-2m$ must be ...
1
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1answer
51 views

If $\alpha,\beta$ are roots of $x^2+px+q=0$ and also of $x^{2n}+p^nx^n+q^n=0$

If $\alpha,\beta$ are roots of $x^2+px+q=0$ and also of $x^{2n}+p^nx^n+q^n=0$ and $\frac{\alpha}{\beta}$,$\frac{\beta}{\alpha}$ are the roots of $x^n+1+(x+1)^n=0$, then $n$ is Odd Even ...
0
votes
2answers
32 views

How to prove a solution of equation is rational if another one is rational number?

The question is : $r$ is the solution of equation $x^2+bx+c=0$ and $r$ is a rational number, so there is another solution $s$, how to prove s is a rational number as well? I have no idea about it and ...
0
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2answers
30 views

Maximums on Quadratic Functions [closed]

How do you find the maximum of a quadratic function? Specifically, $R(x) = -4x^2 + 4000x$
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0answers
14 views

Disadvantages of Taylor series method

There is method called Taylor series method to solve non linear equations iteratively. I am interested to know ,what are the disadvantages of using this method to solve. General Idea any one please?
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2answers
28 views

I don't understand how the following algebraic equation breaks down. I just can't figure out how this answer is devised.

I just don't understand how this equation breaks down like this. The second step... $[k^2 + k + 2k + 1]$ is perplexing, but breaking that down to $(k+1)(k+2)$ has completely baffled me. I would expect ...
4
votes
5answers
69 views

Why is the solution to $\sqrt{6-5x}=x$ only $x=1$ and not $x=-6$? [duplicate]

I solved the equation $\sqrt{6-5x}=x$ as follows: $$(\sqrt{6-5x})^2=x^2$$ $$6-5x=x^2$$ $$0=x^2+5x-6=(x+6)(x-1)$$ $$x=-6 \quad \text{or} \quad x=1$$ If I plug in $x=-6$ into the original equation, I ...
0
votes
1answer
37 views

How to find $f(2)+f^{-1}(5)$ if $f(2x^2+3x+4)=6x^2+9x+20$? [closed]

$$f(2x^2+3x+4)=6x^2+9x+20$$ How to solve $f(2)+f^{-1}(5)$ ? Any help or advice on solving is much appreciated. Thanks!
4
votes
2answers
40 views

Sign of determinant of a $3 \times 3$ matrix with entries $1+\alpha^{i+j-2}+\beta^{i+j-2}$, for distinct $\alpha,\beta\in\mathbb R\setminus\{1\}$

Let $ \alpha\ne1,\beta\ne1$ be the distinct real roots of the equation $$ax^2+bx+c=0,~~a,b,c\in \mathbb{R},a\ne 0$$ Let $S_n=\alpha^n+\beta^n,n\geq0$ and ...
0
votes
1answer
27 views

The nature of roots of the quadratic equation $ax^2+(b-c)x-2b-c-a=0,$

The expression $ax^2+2bx+c$ where $a$ is a non-zero real number,has the same sign as that of $a$,for every real value of $x$,then roots of the quadratic equation $ax^2+(b-c)x-2b-c-a=0,$are $(A)$real ...
0
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1answer
38 views

two positive real numbers have their sum, product… [closed]

Two positive real numbers have their sum,product and difference of the squares $(a^2-b^2)$ equal. Find those numbers. It would be easy to solve if only two of these were mentioned, but I don't know ...
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votes
2answers
57 views

is there a triangle with sides $2,3,6$?

Is there a triangle with $a=2, b=3, c=6$? (I know there's not because sum of any two sides has to be greater than the third side) How much do we need to extend these sides to get a right triangle ...
1
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3answers
30 views

solving rectangle

Diagonal of a rectangle is $13$ cm. If we extend the length of the rectangle for $4$ cm and width for $7$ cm, then diagonal will be longer for $7$ cm as well. Find sides (length and width) of the ...
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2answers
27 views

speed and travel problems

The ship left the harbor and is travelling $60$ kilometers downstream. Then it continues on the tributary river (upstream) $20$ kilometers. Travel took $7$ hours to finish. River speed is $1$ km/h. ...
1
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1answer
19 views

Roots of polynomials combined with Trigonometric Functions

If $$ f(x) = x^2 + ax + d \cos x $$, where $a$ is an integer and $d$ is a real number, what are all possible values of the tuple $(a,d)$ such that $f(x)$ and $f(f(x))$ have the same set of real roots? ...
0
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2answers
29 views

How to derive in a quadratic equation. [closed]

I was reading Purplemath's lesson about quadratic equations, and came to the part about deriving the solution to $ x^2 + 6x + 10 = 0$. I understood the part about putting the loose number in the other ...
1
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3answers
60 views

Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

I have a question. Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$ I don't know how to proceed. ...
0
votes
3answers
51 views

What comes first here? pemdas doesnt really tell me what to do here

So I have this equation: $2x(x+3)(x+3)$ Do I FOIL the $(x+3)$ first or multiply the $2x$ to the first $(x+3)$? Would there be a difference? Isn't multiplication commutative?
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1answer
10 views

Writing the function to maximize volume or a cylinder

A rectangular piece of paper is curled into a cylinder with two open circles on each side. The perimeter of the piece of paper is 124 inches. What is a function that could be written to find the ...
2
votes
4answers
76 views

If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$

If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$ Using the condition for common root, $$(3c-5b)(b-3a)=(c-5a)^2$$ ...
1
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1answer
27 views

Triple Simultaneous Equations not resolving the Equation for a Quadratic Function

so I'm doing this math problem for my Calculus I course in college. Here is a screenshot of the problem: Graph 1 (click here to view); the prompt is "Find an expression for the quadratic function ...
4
votes
3answers
58 views

Is this quadratic pointing up or down? How do I know?

The equation is $-2x^2 + 4x + 30 = 0$. I simplified it to $-2(x^2 - 2x - 15)$. To know if it points up, I need to look at $ax^2$, and if $a > 0$ it is up and if $a$ is $< 0$ it is down. ...
1
vote
3answers
29 views

Which form of this quadratic do i use to solve intercept and range?

So my equation is: $-2x^2 + 4x + 30 = 0$ If I use this form to look at my y intercept, it will be 30. However, once I simplify it to: $x^2 - 2x - 15$, then my y intercept will be $-15$. Which one do ...
0
votes
1answer
36 views

Determine the equation of a parabola with roots $2 + \sqrt {3}$ and $2 - \sqrt {3}$, and passing through the point $(2,5)$

My attempt: $$f(x) = a(x - r)(x - s)$$ $$f(x) = a(x-(2 + \sqrt {3}))(x-(2- \sqrt {3}))$$ From here, I'm stuck. I can't remember where to go from this point and need some help.
2
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1answer
45 views

Writing an equation from data

Wendy, Elizabeth, and Charlie are all working on a math problem together and they are having a disagreement.: Ticket lines are huge at the Math Olympics ticket office. Pi, the local math team, is ...
0
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1answer
21 views

Why Non Linear equations put equal to zero in Newton Raphson Mehotd

While solving non linear equations we put them equal to zero in Newton-Raphson Method.Why we do that? Any Idea?
2
votes
2answers
84 views

Are $\mathbb{C}^2$ and $\mathbb{C}^2/(x,y)\sim(y,x)$ homeomorphic?

Let $A$ be the set of monic quadratics over $\mathbb C$ and let $B$ be the set of unordered pairs over $\mathbb C$ where possibly the two elements of the pair may be the same. Then the map which takes ...
2
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2answers
32 views

Solving a radical equation with trinomials on both sides

$$8\sqrt{a^2-4a-16}=3a^2-12a-64$$ I do know the standard procedure—square both sides, isolate square root, square again, check solutions to make sure they are real, etc. However, for a problem such ...
0
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1answer
30 views

Solve equation for t

$$s = 2 \ln|\tan(t) + \sec(t)|$$ I tried to solve it and got a quadratic equation which turned out to equal $arcsin(\dfrac{-2 \pm e^s}{2(1+e^s)})$ This doesn't seem right. Any thoughts?
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3answers
65 views

Find all rational points where $x^2 - y^2 = 1$ (need help simplifying quadratic formula) [duplicate]

The original problem is to find all rational points where $x^2 - y^2 = 1$ I know how to go about the problem, but whenever I get to the point of simplifying my equation, I keep having problems. This ...
1
vote
1answer
25 views

Quadratic function that produces natural number from natural number inputs

I am currently trying to find a way to generate different (preferably quadratic) function as part of a encryption algorithm such that : ...
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2answers
41 views

Roots of $f(x)+g(x)$

Question : Let $p,q,r,s \in \mathbb R$ such that $pr=2(q+s)$. Show that either $f(x)=x^2+px+q=0$ or $g(x)=x^2+rx+s=0$ has real roots . My method : To the contrary suppose that both $f(x)$ and ...
0
votes
3answers
53 views

Why not always use the quadratic equation

The is a very simple question, but I have just started studying quadratics. I understand how to factor them using different methods and also understand solving a quadratic using the formula, but my ...
1
vote
1answer
43 views

Find integers $x$ and $y$ such that $\frac{27^{x+y}}{9^{xy}}=27$ and $\frac{4^{2xy}}{8^{x+y}}=512$ .

Find all the integers $x$ and $y$ such that : $$\frac{27^{x+y}}{9^{xy}}=27$$ and :$$\frac{4^{2xy}}{8^{x+y}}=512$$ I'm in Algebra two and I feel like there are certain types of math I haven't ...
2
votes
3answers
76 views

How can you find $m$ in $mx^2+(m-3)x+1=0 $ so that there is only one solution

How can you find $m$ in $$mx^2+(m-3)x+1=0 $$ so that there is only one solution. I tried to solve it by quadratic equation but I end up with two solutions. So I want it know that is there a way so ...