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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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3answers
23 views

How do I prove that $a^2 + b^2$ is not a prime number from this given quadratic equation?

Given the quadratic equation $$x^2 + ax + b + 1 = 0$$ Where a, b are integers, and the roots are natural number. Prove that $a^2 + b^2$ is not a prime number. What should I do to prove it? Can anyone ...
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0answers
21 views

Find the value of $(a, b, c)$ with $a, b, c \in \Bbb N$

$$x^2 - 2ax + b = 0$$ $$x^2 - 2bx + c = 0$$ $$x^2 - 2cx + a = 0$$ If all the roots of all three equations are a natural number, what is the value of a, b, and c?
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4answers
36 views

If $\sqrt{27-10\sqrt{2}} = a+b$, where $a$ is a positive integer and $b$ is between $0$ and $1$, what is $\frac{a+b}{a-b}$?

If $\sqrt{27-10\sqrt{2}} = a+b$, where $a$ is a positive integer and $b$ is between $0$ and $1$, what is $\frac{a+b}{a-b}$? I actually have no idea how to start this question, other than to square ...
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0answers
24 views

How do I express “x” from this equation?: $y=x/(x^2+1)$ [duplicate]

How do I express "x" from this equation?: $y=x/(x^2+1)$. I am not able to express it.. What I really want to do is to find out values of this function: $f(x)=x/(x^2+1)$ if the domain is R{1,-1}. I ...
2
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3answers
64 views

For which values of $k$ does the equation $2\cos^{2}\theta +k\sin \theta + k = 2$ have real solutions?

So I take A level maths and this question was in our textbook. We solved an inequality for when the discriminant is less than zero and this gave us the same answer that is in the textbook. The ...
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2answers
57 views

Is the function $f: R \to R$ defined by $y=x^{2}-2x-2$ a surjection?

The problem is to find if the following function is a surjection. $f: R\to R$ defined by $y=x^{2}-2x-2$ I know that it is not a surjection by looking at a graph of the function but I am new to ...
4
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3answers
53 views

Find the condition for the three equations $a_rx^2+b_rx+c_r=0$; $r=1,2,3$ to have a common root.

Find the condition for the three equations $a_rx^2+b_rx+c_r=0$; $r=1,2,3$ to have a common root. My attempt is as follows: \begin{equation} a_1x^2+b_1x+c_1=0\tag{1} \end{equation} \begin{equation} ...
1
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2answers
36 views

If each pair of equations $x^2=b_1x+c_1=0,x^2=b_2x+c_2 \text{ and } x^2+b_3x=c_3$ have a common root, prove following

If each pair of equations $x^2=b_1x+c_1=0,x^2=b_2x+c_2 \text{ and } x^2+b_3x=c_3$ have a common root, prove that $(b_1+b_2+b_3)^2=4(c_1+c_2+c_3+b_1b_2)$ My attempt is as follows: For equations $x^2=...
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1answer
41 views

If $x^5-bx^3+cx^2+dx-e$ can be expressed as the product of a perfect square and a perfect cube then prove following

If $x^5-bx^3+cx^2+dx-e$ can be expressed as the product of a perfect square and a perfect cube then prove that $$\frac{12b}{5}=\frac{9d}{b}=\frac{5e}{c}=\frac{d^2}{c^2}$$ My attempt is as follows: $...
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4answers
25 views

parallel shortest distance between graph of parabola and line [on hold]

I would be really thankful if someone could explain how I can approach this problem. (Not only this one but this type of finding the distance between parabola and line) Problem n32 ECONOMICS ...
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2answers
36 views

If $a^2+c^2>ab$ and $b^2>4c^2$ , for real x, show that $\frac{x+a}{x^2+bx+c^2}$ cannot lie between two limits

If $a^2+c^2>ab$ and $b^2>4c^2$ , for real x, show that $\frac{x+a}{x^2+bx+c^2}$ cannot lie between two limits My attempt is as follows: $$y=\frac{x+a}{x^2+bx+c^2}$$ $$yx^2+byx+yc^2=x+a$$ $$yx^...
2
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3answers
65 views

$\{a^3+(1-\sqrt{2})a^2-(3+\sqrt{2})a+3\sqrt{2}\}x^2+2(a^2-2)x+a>-\sqrt{2}$

If $\{a^3+(1-\sqrt{2})a^2-(3+\sqrt{2})a+3\sqrt{2}\}x^2+2(a^2-2)x+a>-\sqrt{2}$ is satisfied for all real $x>0$ then obtain the possible values of the parameter $a$. My attempt is as follows: $$\...
2
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2answers
60 views

Find $a\in \mathbb{R}$ for which $a\cdot \left(\frac{1}{1+x^2}\right)^2-3\cdot\frac{a}{1+x^2}+1=0$ will have all roots imaginary

Find $a\in \mathbb{R}$ for which $a\cdot \left(\frac{1}{1+x^2}\right)^2-3\cdot\frac{a}{1+x^2}+1=0$ will have all roots imaginary. My attempt is as follows:- Let $t=\frac{1}{1+x^2}$, and let's find ...
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1answer
30 views

Graph of $x^2$ + $y$ $=$ $0$ is an upward or a downward opening parabola?

That is my exact question. If we graph $x^2$ + $y$ $=$ $0$, do we get a downward opening parabola? Let me explain what actually got me confused. I know that $y$ = $x^2$ is an upward opening parabola ...
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2answers
35 views

Find all real values of the parameter a for which the equation $x^4+2ax^3+x^2+2ax+1=0$ has

Find all real values of the parameter a for which the equation $x^4+2ax^3+x^2+2ax+1=0$ has 1) exactly two distinct negative roots 2) at least two distinct negative roots I tried to factorize it but ...
2
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1answer
44 views

Solve for k, $f(x)=x^2+2(k-1)x+k+5, k\in R$

If the graph of the function $f(x)=x^2+2(k-1)x+k+5, k\in R$ cut the x-axis at least at one point on the positive side , find the set of possible values of the constant k. My attempt is as follows: ...
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0answers
20 views

Finding a direct function for $f_n = f_{n-1} + (x-a_{n-1})^2$ where $a_{n-1}$ is root of $f_{n-1}$ closest to $\alpha$ and $f_1 = x-a$.

Problem Suppose $f_1 = (x-a)$ and $f_n = f_{n-1} + (x-a_{n-1})^2$ where $a_{n-1}$ is a root of $f_{n-1}$ closest to $0$. Is there a function for $f_n$ that only depends on $x$? Examples As a first ...
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1answer
49 views

How to prove that $ B = A^{-1} $ ? If A and B are 2 x 2 matrices where $ B \neq I_2 $ such that $ ( A + B )^2 = A^2 + 2AB + B^2 $

If $A \text{ and } B \text{ are } 2\times 2$ matrices where $B\neq I_2$, such that $(A+B)^2 = A^2 +2AB + B^2$, deduce that $B = A^{-1}$ If $A = \begin{bmatrix}1&2\\9&-1\end{bmatrix}$ ...
14
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4answers
192 views

Find a such that $ax^{17}+bx^{16}+1$ is divisible by $x^2-x-1$.

Find $a$ such that $ax^{17}+bx^{16}+1$ is divisible by $x^2-x-1$. I tried taking the roots of the polynomial which are $\frac{1±\sqrt{5}}{2}$ And I got the equation $a(\frac{1±\sqrt{5}}{2})^{17}+b(\...
6
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1answer
40 views

The polynomial $x^{2k}+1+(x+1)^{2k}$ is not divisible by $x^2+x+1$. Find the value of $k$ belongs to $\mathbb N$.

The polynomial $x^{2k}+1+(x+1)^{2k}$ is not divisible by $x^2+x+1$. Find the value of $k\in \mathbb{N}$. I tried finding out the roots of $x^2+x+1$ which were $\frac{-1±\sqrt{3}i}{2}$ but in vain. I ...
2
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5answers
88 views

Find the number of real solutions of the equation $2^x+x^2=1$

My attempt is as follows:- $$2^x+x^2=1$$ $$\left(1+x\cdot log(2)+\frac{x^2\cdot (log(2))^2}{2!}+\frac{x^3\cdot (log(2))^3}{3!}+\dots\right)+x^2=1$$ $$x\cdot log(2)\left(1+x^2+\frac{x\cdot log(2)}{2!}+...
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1answer
31 views

Solve for x for the following description

Let $x$ be a perfect square and a natural number. When $x$ is divided by $5$, the quotient is $[x]$ and the remainder is $\{x\}$. Then solve for x if $$\sqrt{x}+\{x\}=[x]$$ My attempt is as follows: ...
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3answers
31 views

Find the range of values of $k$ for which $kx^2 + 8x + k <6$ for all real values of $k$

Find the range of values of $k$ for which $kx^2 + 8x + k <6 $ for all real values of $k$. I'm unsure if the discriminant must be greater than zero or less than zero. My working steps: \begin{...
1
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2answers
41 views

factorization in proof by induction

I have to prove that Sum of series $$\sum_{k=1}^n k^2 = \frac{(n)(n+1)(2n+1)}{6}$$ I tried doing it myself, but was not able to factorise the third grade polinomial. So I watched this youtube video, ...
3
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0answers
21 views

Form the equation of lowest degree with rational coefficients which has…?

Form the equation of the lowest degree with rational coefficients which has $2+\sqrt{3}$ and $3+\sqrt{2}$ as two of its roots I tried taking $x=2+\sqrt{3}$, $3+\sqrt{2}$ and also $y=\sqrt{3}$, $y=\...
1
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4answers
37 views

Find $x,y,z$ for the given conditions

$$4x^2+25y^2+9z^2-10xy-15yz-6zx=0$$ $$x+y+z=5$$ I tried two approaches 1) Substituting $x$ as $5-y-z$ in the first equation but didn't work out, I was getting $39y^2+19z^2-31yz-90y-70z+100=0$ ...
4
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2answers
52 views

Find the polynomial equation of the lowest degree with rational coefficients whose one root is…?

Find the polynomial equation of the lowest degree with rational coefficients whose one root is $\sqrt[3]{2}+3\sqrt[3]{4}$ I tried using the conjugate pairs but I couldn't solve it for any polynomial ...
0
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4answers
34 views

When does $ 9k^2 (x-5)^2 - 125k^2 \geq (9+5k^2)(x^2 - 10x) + 225 $ have a unique solution

$$ 9k^2 (x-5)^2 - 125k^2 \geq (9+5k^2)(x^2 - 10x) + 225 $$ For which value of the constant k below will the inequality have a unique solution? choices are: $1/2014, 3/2, -9, 2014$ I have already ...
3
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1answer
58 views

How might I define a parabola in vertex form, such that…

Given the formula for a parabola in vertex form: $y = a(x-h)^2+k$, as $h$ and $k$ are changed, the $a$ value will adjust in order to keep the left hand $x$-intercept anchored to the origin. I'm really ...
1
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1answer
20 views

Interval that the sum of the roots of a quadratic equation can be in.

I am given the equation: $2x^2-2mx+m^2-2m=0$ Where $m\in \mathbb{R}$; $x_1$ and $x_2$ are the roots of the equation. The question asks for the interval of the sum of the roots, that is $x_1 + x_2$. ...
2
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2answers
35 views

Find the condition when exactly one root of given quadratic equation lies in the given interval

Let $f(x)=x^2-2px+p^2-1$, then find the condition when exactly one root of given quadratic equation lies in the interval $\left(-2,4\right)$ My attempt is as follows: $$x^2-2px+p^2-1=0$$ $$\left(x-p\...
1
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3answers
26 views

Find all the values of the parameter 'a' for which the given inequality is satisfied for all real values of x.

Find all the values of the parameter 'a' for which the inequality is satisfied for all real values of x. $$a\cdot 9^x+4\cdot \left(a-1\right)\cdot 3^x+\left(a-1\right)>0$$ My attempt is as ...
1
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3answers
39 views

Domain of inequation inside squared root

How should I go when restricting the roots of the inequation: $\sqrt {x^2+5x+6} - \sqrt {x^2-x+1} \lt 1$? By restricting both the squared roots, I know that: $x \le 3$ and $x \ge -2$ However when ...
2
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1answer
35 views

Help solving system of equations

I've been trying to solve this problem for a good 2 hours and I still couldn't find the solution please help. $$ \left\{ \begin{array}{c} 4x^2-3xy-y^2=0 \\ 32x^2-36xy+9y^2=6 \\ \end{array} \right. ...
0
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4answers
67 views

Given $x^2+y^2>0$ where $x,y \in R$ [closed]

Find the maximum and minimum value of the following expression E $$E=\dfrac{x^2+y^2}{x^2+xy+4y^2}$$
1
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2answers
28 views

Factoring a quadratic

On page 7 of “Elliptic Tales: curves, counting and number theory” the authors state that: $(m^2+1)x^2+(2mb)x+(b^2-1)=0$ Has the factors: $(m^2+1)(x-\alpha)(x-\beta)=0$ Where $\alpha$ and $\beta$ ...
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3answers
41 views

A free website which can solve this equation

Am looking for a website which can solve quadratic equations such this solve for $v$ , $h(v-t) = h(v+t) $ where $h(x) = ax^2 + bx +c$. The value is the vertex of the quadratic function the website ...
0
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3answers
31 views

Quadratic equation roots

How do I get the roots of the quadratic equation? $dx^2+(d ^2−d+1)x+d−1 = 0$ The solution is supposed to be $1-d$ and $-1/d$ but I have no idea how to get to it.
1
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3answers
50 views

Consider parabola $x^2 + Bx$, as $B$ varies why does the vertex trace out a parabola? Would love Geometric Intuition

I have been playing around with desmos and found this strange idea (I encourage you to view here to visualise what I am talking about: https://www.desmos.com/calculator/dwqp8jmp7v) In high school I ...
2
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1answer
69 views

The value of $P^2+Q^2$ given that the reciprocals of the roots of two polynomials follow an arithmetic progression.

If $\alpha,\gamma$ are non zero roots of the equation $Px^2-4x+1=0$ and $\beta,\delta$ are non zero roots of $Qx^2-6x+1=0$ (where $\frac{1}{\alpha}$, $\frac{1}{\beta}$, $\frac{1}{\gamma}$ and $\frac{1}...
0
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0answers
9 views

Quadratic model for supply and demand

For the following supply and demand functions, p denotes the unit price in dollars and x denotes the number of units supplied or demanded of some product in millions of units. $$Supply: p = x^2 + 8$$ $...
-1
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1answer
28 views

Golden Mean of Rectangle

ABCD is a rectangle with length and breadth in the ratio α : 1. It is divided into a square APQD and a second rectangle PBCQ, as shown. Show that the length and breadth of rectangle PBCQ are also ...
2
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2answers
59 views

If the difference between the roots of the equation $x^2+a\cdot x+1=0$ is less than $\sqrt{5}$, then find the set of possible values of a.

If the difference between the roots of the equation $x^2+a\cdot x+1=0$ is less than $\sqrt{5}$, then find the set of possible values of a. My attempt is as follows: $$\left|\frac{\sqrt{D}}{1}\right|&...
3
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1answer
63 views

A Question on Quadratic Equations.

Show that the expression $\frac{(ax-b)(dx-c)}{(bx-a)(cx-d)}\\$ will be capable of all values when x is real, if $a^2-b^2$ and $c^2-d^2$ have the same sign. Here's my approach: I tried equating it ...
1
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2answers
60 views

Is this system of equations “solvable”?

I've been trying to solve the following system of equations for $p_1>0$ and $p_2>0$, but have failed to do so (in a reasonable fashion). Can anybody confirm that this is indeed not possible or ...
1
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2answers
65 views

Solve the equation $\sqrt{a(2^{x}-2)+1}=1-2^{x}$ for every value of the parameter $a$.

Question : Solve the equation $\sqrt{a(2^{x}-2)+1}=1-2^{x}$ for every value of the parameter a. I have solved the problem as follows $\sqrt{a\left(2^{x}-2\right)+1}=1-2^{x}$ $a\left(2^{x}-2\right)+...
1
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1answer
22 views

quadratic equation modulo product of coprime squarefree integers

Let $a, b, c$, pairwise coprime squarefree integers. Suppose $au^2 + bv^2 + cw^2 ≡ 0 (mod\ |abc|)$ with $au^2 , bv^2 , cw^2$ pairwise coprime. Prove that if $(x,y,z) \in Λ_0 := \{(x, y, z) ⊂ \Bbb Z^...
0
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2answers
39 views

Proving that the quadratic equation $(q-5)x^2 +5x -q=0$ has real roots for any value of $q$

Prove that the quadratic equation $(q-5)x^2 +5x -q=0$ has real roots for any value of $q$. So I have already tried using the discriminant but just wanted to see if my answer is right or not. ...
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1answer
35 views

Partial fractions integration - Constant denominator value

How do I chose which factor goes under what constant? I understand that $x^{2}+x = Ax+B$ or $Bx+C$ etc (That is not my question/concern) My question is: When I factor a quadratic, say for example: ...
2
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1answer
40 views

is the quadratic equation appropriate for this?

I work in a paper mill as a tech. There is a formula for percent solvents in a liquor solution. It is s=(A*P^2) + (B*P) $S$ is the percent solvent, $A$ and $B$ are ...