Questions tagged [quadratics]

Questions about 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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Quadratic With Periodic Coefficents

I've come across a problem that results in the equation: $t -2t\sin t -2\cos t -2 = 0$ I've tried to do this analytically but I can't figure it out. At this point, I just want to know if it's even ...
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If a downwards curve's tangent point is at $(0, -15)$, what is the equation for the tangent line of this curve?

Question is in the title. If a downwards curve's tangent point is at $(0, -15)$. So, what is the equation for the tangent line of this curve? Edit: the equation of the curve is $y = x^2 + 2x - 15$
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2 votes
1 answer
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Multiplying by $1$ adds a solution to an equation

I have a question, which is motivated by my book's solution to finding the inverse function of $f(x)=\frac{x}{1-x^2}$ with the domain of $f(x)$ restricted the open interval $(-1,1)$. The questions are ...
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Check Understanding of $\varepsilon$-$\delta$ Limit Proof

Say, we want to prove that $\lim_{x \to a} x^2 = a^2\;$ Assuming $a>0$ here. Here’s how I would think the $\varepsilon$-$\delta$ proof way. Please give feedback on thinking. $$\forall \varepsilon&...
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-2 votes
1 answer
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Can $f(x) = \frac1x$ be expressed in quadratic form for $x > 0$?

Problem Can $f(x) = \dfrac1x$ be expressed in quadratic form for $x > 0\space?$ My Approach I think it might be possible, unless I misunderstand something about the properties of $f(x)$. We can ...
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-2 votes
0 answers
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Solving quadratic inequality via number line [closed]

i Need to solve a quadratic inequality and the only information I’m given are two points on a number line The points I’m given are -2 (including) and 3 (including). I need to solve for a, b, and c in ...
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1 answer
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How come discriminant can be used to find range of a function.

Find range of function $f(x)=x^2+6x-1$ for domain $x \in \mathbb{R}$. The solution is: Let $y=x^2+6x-1$ then $0=x^2+6x-1-y$ comparing with $0=ax^2+bx+c$ gives $a=1,b=6,c=-(1+y)$ For real $x$, $b^2-4ac ...
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1 vote
1 answer
67 views

Find $ k $ if both roots of this equation $ x^2-2kx+k^2+k-5=0 $ are less than 5

If both roots of this equation $ x^2-2kx+k^2+k-5=0 $ are less than $ 5$, then find the limit of $ k$. Method-1 Let's say, \begin{align} f(x)=x^2-2kx+k^2+k-5 \end{align} \begin{align} f(0)=0 →\alpha,\...
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3 answers
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Forming equation from given roots ($ -\alpha, -\beta $) where $ \alpha,\beta$ are the two roots of $ \ ax^2+bx+c=0$

If this equation $ \ ax^2+bx+c=0 $ has two roots $ \alpha,\beta$ then form an equation which has the roots $ -\alpha,-\beta $ Solution (given): Here, \begin{align} ax^2+bx+c=0 →\alpha,\beta \end{align}...
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Understanding the strategy used for selecting the f(x) values to get the accurate maxima which is being attained too

Prove that $|a|+|b|+|c|\le17$ if $|ax^2+bx+c|\le1$ for $0\le x\le1$ with reference to the above stack post where a solution was given to it . I have fully understood the method , but would like to ...
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2 votes
2 answers
54 views

Find equation of a parabola when the area cut by a straight line and two intersection points are given.

I have equation of a straight line $y = -\frac{0.45}{0.248}x - 1.1330645$ It intersects a quadratic function graph (parabloa) at two points (-0.652, 0.05) & (-0.9, 0.5) I have the area bounded ...
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14 votes
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187 views

Does the limit of the cubic formula approach the quadratic one as the cubic coefficient goes to $0$?

The formula for solving a cubic equation of the form $ax^3+bx^2+cx+d=0$ does not seem to yield the quadratic formula for the limit $\lim _{a \rightarrow 0} \text{(cubic formula)}$. But, if one tries ...
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1 vote
1 answer
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Necessary and sufficient condition for roots of quadratic with complex coefficient lie within unit disc centred origin.

Let $z_1,z_2$ be the roots of $az^2+bz+c=0;\,\,a,b,c\in \mathbb{C},\,\,a\ne 0$. Find necessary and sufficient condition for which $\max\{|z_1|,|z_2|\}<1$. My progress: $z_1+z_2=-\frac ba\implies |b|...
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Let $z_1$ and $z_2$ be the roots of $az^2+bz+c=0;\,a,b,c\in \mathbb{C}, a\ne 0$ and $w_1,w_2$ be roots of $(a+\bar{c})z^2+(b+\bar{b})z+(\bar{a}+c)=0$

Let $z_1$ and $z_2$ be the roots of $az^2+bz+c=0;\,\,a,b,c\in \mathbb{C}, a\ne 0$ and $w_1,w_2$ be roots of $(a+\overline{c})z^2+(b+\overline{b})z+(\overline{a}+c)=0$. If $|z_1|<1,|z_2|<1$, then ...
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3 votes
2 answers
66 views

Range of the function $f(x) = 4^x + 4^{-x} + 2^x + 2^{-x} + 3$

I tried to form a quadratic equation by taking $2^x = u$ $$u^2 + \frac{1}{u^2}+u + \frac{1}{u}+3$$ $$\left(u + \frac{1}{u}\right)^2+u + \frac{1}{u}+1$$ taking $u+1/u = t$ $$t^2+t+1$$ From this ...
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Solving a Ricatti type equation

Could anyone help me solve this equation? Let $n,m,l$ be positive integers such that $n \ge m$ and $l>m$, but there is no such constraint between $l$ and $n$. We have two known matrices $Q_1 \in \...
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0 votes
1 answer
101 views

A pattern that leads to regular continued fractions of quadratic irrationals [closed]

The following expression can be obtained by converting the continued fraction of quadratic irrationals to single fraction. $$ \sqrt{N} = \frac{b\sqrt{N}+aN}{a\sqrt{N}+b} $$ The equation holds for any ...
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2 votes
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Why transposing doesn't work (is not working) in quadratic equation?

For example I have an equation: $4k^2-24k+0=0$ $\implies 4k^2 = 24k$ $\implies 4 × k × k = 24 × k$ $\implies 4k = 24$ $\implies k = 6$ But it doesn't find $0$ as an answer that quadratic equation ...
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0 votes
1 answer
33 views

Quadratic formula- calculator gives an answer but cannot do it manually

Embarrassing but I have this equation: If I use my calculator's quadratic mode I get the correct answer. However, if I try to do it manually I get a problem because 16 - 4 x 4 x 840.15 gives a ...
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0 votes
1 answer
27 views

Rewrite proposition with logical symbols

I want to rewrite the following proposition in mathematical language (and by mathematical language I mean symbols such as: $\forall , \exists, (, ), \implies$ and so on). Proposition: Every non-...
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30 views

Optimization doubt

Let $f(x)=x^2+ax+b$ is a function $a,b \in \mathbb R, a^2>4b$ and $g(x)=x^2+2x-1 $ such that $f(g(x_i))=0\ \forall\ i\in \{ 1,2,3,4\}$ and $x_i<x_i+1\ \forall \ i\in \{1,2,3 \}$ and $x_1,x_2,...
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0 votes
1 answer
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When is $n = \frac{a_{1}x^{2} + a_{2}x + a_{3}}{a_{4}x + a_{5}}$ an integer where $a_{i}$ are incredibly large integers? [closed]

I have the equation with the following form: $$n = \frac{a_{1}x^{2} + a_{2}x + a_{3}}{a_{4}x + a_{5}}$$ Where $a_{i}$ are incredibly large (e.g 1000 digits) but unrelated (not part of sequence or ...
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2 votes
3 answers
95 views

When is $K = an^2 + bn + c$ a square number?

Suppose I had the equation: $K = an^2 + bn + c$ where: $n$ is a positive unknown integer. $a,b,c$ are positive known integers. Problem: What values of $n$ make $K$ a square number? (1a) Is there any ...
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Show convexity of a quadratic function

Given quadratic function $f : \mathbb{R}^n \to \mathbb{R}$ defined by $f(\mathbf{x})=\mathbf{x}^\top \mathbf{A} \, \mathbf{x}+b$ where the matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$ is symmetric ...
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1 vote
0 answers
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Why is there a term $3.70074\times 10^{-17}x^2$ in this solution?

I was using WolframAlpha to find a polynomial of degree up to $2$ that best fits the points $(-1,2),(0,0),(1,-2),(2,-4)$. The answer I was expecting is $$0x^2-2x+0.$$ However, here is what ...
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How to solve the matrix equation $\mbox{tr}(\mathbf{X}\mathbf{X}^H)\overline{\mathbf{B}}=\mbox{tr}(\mathbf{X}\mathbf{X}^H\mathbf{B})\mathbf{I}$?

I want to solve the following equation for $\mathbf{X}\in\mathbb{C}^{N\times M}$, with $M < N$: $$\mbox{tr}(\mathbf{X}\mathbf{X}^H)\overline{\mathbf{B}}=\mbox{tr}(\mathbf{X}\mathbf{X}^H\mathbf{B})\...
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1 vote
2 answers
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Mistake in another approach for the old problem

In the question being discussed here Prove that $|f(x)| \le \frac{3}{2}$ when $f(x)=ax^2+bx+c$ i get it how they used the triangle inequality and the bound here that is this : $$|f(x)|=|(\frac{f(1)}{...
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Maximum value of f(x) in the given domain according to the given constraints: [duplicate]

Let $f(x) =ax^2 +bx+c,$ where $a, b ,c\in\mathbb{R}.$ If $f(-1), f(0) , f(1)\in [-1,1]. $ Then prove that $|f(x) |\le\frac{3}{2}\forall x\in [-1,1]$. My method was involving graph : first of all ...
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1 answer
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Proving that a quadratic equation has no integer solution if c=2n and n is odd

My textbook makes the proposition: For all integers $m$ and $n$, if $n$ is odd, then the equation \begin{align} x^2+2mx+2n=0 \end{align} Has no integer solution for $x$. It asks me to prove via a ...
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1 vote
1 answer
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This questions is related to the signs of the coefficients of quadratic polynomials.however doesn't ask about the coefficients themselves.

Suppose a,b,c are three real numbers, such that the quadratic equation x²-(a+b+c)x+(ab+bc+ac)=0,has roots of the form α±iβ, where α>0 β≠0 are real numbers (i=sqrt(-1)). Show that (I) the numbers a,...
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This is question about the integral solutions of 2 different quadratic equations(Question from INMO 1995) [duplicate]

Show that there are infinitely many pairs(a,b) of relatively prime integers (not necessarily positive) such that both quadratic equations x²+ax+b=0 and x²+2ax+b=0 have integer roots I have no idea on ...
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1 vote
1 answer
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This question is asking about the number of quadratic polynomials possible under given conditions [closed]

Find the number of quadratic polynomials ax²+bx+c ,which satisfy the following conditions : (I) a,b,c are distinct (II) a,b,c ∈ {1,2,3,4......999} (III) (x+1) divides (ax²+bx+c) Solution by me: Since ...
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-3 votes
3 answers
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If both the roots of the equation $x^2 - (m+3)x -2m = 0$ are positive and distinct, where m is a real number, then the correct option is? [closed]

The three options a) m < -3 b) m > 3 c) 2√10 -7 <m < 0 d) m > 2√10 + 7 I did try to solve using quadratic formula but I really can't find the value ...
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1 vote
1 answer
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Hessian of quadratic objective function

I have the quadratic function $$f(\boldsymbol{x}) = \frac{3}{2} \left (x_{1}^{2}+x_{2}^{2} \right) + (1+a) x_{1} x_{2} - \left(x_{1} + x_{2} \right) + b$$ where $a, b \in \Bbb R$ are unknown ...
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2 votes
1 answer
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If $P_n=\alpha^n+\beta^n\;, \alpha+\beta=1, \;\alpha \cdot \beta=-1,\;P_{n-1}=11,\; P_{n+1}=29$ Find $(P_n)^2,\;$ where $n\in \mathbb N$

If $P_n=\alpha^n+\beta^n\;, \alpha+\beta=1, \;\alpha \cdot \beta=-1,\;P_{n-1}=11,\; P_{n+1}=29$, $\alpha$ and $\beta$ are real numbers. Find $(P_n)^2,\;$ where $n\in \mathbb N$ My Approach: Method $1$...
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0 votes
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The problem of minimizing area on the hypotenuse

Consider a right angled triangle whose perimeter is $30 cm$. Suppose the hypotenuse side has length $13 cm$. It is clear that the area of the square on the hypotenuse side is $169cm^2$. I now find out ...
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1 vote
3 answers
60 views

Quadratic equation: understanding how the absolute values in the derivation correspond to the $\pm$ symbol in the classic quadratic formula expression

I'd like if someone could help me understand the typical form of the quadratic formula, which, for the equation $ax^2+bx+c=0$, reads as $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $x \in \mathbb R$ ...
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0 votes
0 answers
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How to solve a quadratic Diophantine equation without trial and error by just inputting integer values for one of the variables?

I’m looking for a way to find only integer solution pairs to a dual-variable quadratic equation without trial and error. For example: $$(a+3\sqrt 5)^2+a-b\sqrt 5=51$$ Valid solution pairs are any ...
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2 answers
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For $\alpha, \beta$ the roots of $x^2-x-1=0$, define $a_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ and $b_n=a_{n-1}+a_{n+1}$. Then ...

Let $\alpha$ and $\beta$ be the roots of $x^{2}-x-1=0$, with $\alpha>\beta$. For all positive integers $n$, define $a_{n}=\frac{a^{n}-\beta^{n}}{\alpha-\beta}, \quad n \geq 1, b_{1}=1$ and $b_{n}=...
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2 votes
0 answers
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A quadratic function $f(x)$ satisfies the inequality $-1 < f(x) < 1$ for $x \in [0, 1]$. What can we say about the range of its coefficients?

Let a function $f(x) = ax^2 + bx + c$, where $a, b, c \in R$, satisfy $-1 \leq f(x) \leq 1$ for all $x \in [0, 1]$ then which of the following conclusions can be made? A) $|a| \leq 8$ B) $|b| \leq 8$ ...
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5 votes
1 answer
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If $|ax^2+bx+c|\leq 2\ \ \forall x\in[-1,1]$ then find the maximum value of $|cx^2+2bx+4a|\ \ \forall x\in [-2,2]$.

If $$\left|ax^2+bx+c\right|\leq 2\quad \forall x\in[-1,1]$$ then find the maximum value of $$\left|cx^2+2bx+4a\right|\quad \forall x\in [-2,2].$$ My Attempt Let $f(x)=ax^2+bx+c$, then $$|cx^2+2bx+4a|=...
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2 votes
1 answer
52 views

What's the value of a quadratic function when $x=0$ if the degree of the constant term is $0$? [duplicate]

The degree of the constant term in a quadratic (or any polynomial) is $0$. Say, I have the following quadratic function: $$f(x) = 2x^2 + 4x +7$$ Since the degree of the constant term is $0$, I can ...
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0 votes
1 answer
21 views

Discriminant involving variance-covariance matrices

I'm trying to show a quadratic equation of the form $$\frac{1}{2} a' \Sigma^{-1} a x^2 - a' \Sigma^{-1} \iota x + \frac{1}{2} \iota' \Sigma^{-1} \iota$$ has a real solution. Here $\Sigma$ is an $N \...
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1 vote
1 answer
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formula for coefficients of Least Squares Regression Quad?

According to Maths Is Fun, the formula for coefficients of Least Squares Regression Line is: ...
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-3 votes
1 answer
46 views

quadratic expression problem [closed]

I was going through my old mock exams I found this question wich I still dont know how to solve. I've done some working out, however I don't really understand the format of the answer sheet plus don't ...
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1 vote
1 answer
18 views

Determining the signs of certain expressions formed by the coefficients of a quadratic (Graph of quadratic given)

Consider the quadratic $y=ax^2+bx+c$ with the following graph: I'm trying to figure out the sign of the following expressions: 1). $\frac{c}{a}$ 2). $b+4a$ 3). $2a+b$ The vertex form may prove to be ...
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2 votes
0 answers
55 views

Condition for two quadratic equation to have one common root (Simplification)

If a,b,c are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx2+2ex+f=0$ have a common root if $\frac da, \frac eb, \frac fc$ are in: Arithmetic Progression Geometric Progression ...
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0 votes
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To find Possible values of combinations of a & b for $a^2 + b^2 = 576$ or any quadratic equation under certain restrictions.

Q: Find possible values of combination of a&b such that $a^2 + b^2 = 576$ or any other quadratic equation under certain restrictions. The sole purpose is to understand how can we do it: CONDITION: ...
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2 votes
3 answers
126 views

Finding the slope of line intersecting the parabola

A line $y=mx+c$ intersects the parabola $y=x^2$ at points $A$ and $B$. The line $AB$ intersects the $y$-axis at point $P$. If $AP−BP=1$, then find $m^2$. where $m > 0$. so far I know $x^2−mx−c=0,$ ...
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0 votes
2 answers
51 views

Substituting in variable at earlier stage in equation

I have a somewhat very basic question. I have the following equations. Eq1 is the first raw form and then further simplying it leads to Eq2 My question is. If I substitute k=0 in Eq1, I get 0 = -1 ...
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