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 $y=a(x+b)(x+c)$.

2
votes
2answers
64 views

The roots of the equation $z^2+pz+q=0,$ where $p,q$ are complex numbers, are ..

I am stuck with the following problem that says: The roots of the equation $z^2+pz+q=0,$ where $p,q$ are complex numbers, are represented by the points $A,B$ on the complex plane. If $OA=OB$ and $...
2
votes
2answers
77 views

Question from the 2011 IMC (International Mathematics Competition) Key Stage III paper, about the evaluation of a quadratic equation

When $a=1, 2, 3, ..., 2010, 2011$, the roots of the equation $x^2-2x-a^2-a=0$ are $(a_1, b_1), (a_2, b_2), (a_3, b_3),\cdots, (a_{2010}, b_{2010}), (a_{2011}, b_{2011})$ respectively. Evaluate: $...
0
votes
1answer
70 views

What is the Hessian w.r.t. to matrix X of this quadratic function?

I am stuck in finding the Hessian w.r.t. to matrix $X \in R^{m \times n}$ in the following : $$\frac{1}{2} ||AXB-C||_F^2$$ where $A \in R^{l \times m}$ and $B \in R^{n \times o}$ I got the first ...
2
votes
3answers
34 views

Function with 2 unknowns and one needs to be solved [closed]

I have a homework question stating: Find the set of values of $k$ for which $f(x)=3x^2-5x-k>1$ for all $x\in\mathbb R$. This question has really confused me because I looked at the answer and ...
0
votes
0answers
69 views

How to deal with $x^T(A^TA+B^TB)x$?

I want to minimize $$\|Ax-b\|_2^2 + \|Cx-d\|_2^2 + \|x\|_1$$ I know we can introduce an auxiliary variable and use ADMM to separate the $\ell_2$ and $\ell_1$ norms. But is there a more convenient ...
0
votes
0answers
24 views

Generating function with multiple variables with perfect square

Consider trinomial $x^2 + bx + c = 0$. When $b = 205$ and $c = -206$, then $x = 1$ and the function evaluates to a perfect square. My question is, how can I choose $b$ and $c$ and generate functions ...
0
votes
3answers
40 views

$3x^2+5xy-2y^2-3x+8y+\lambda$ How to express as two factors

$3x^2+5xy-2y^2-3x+8y+\lambda$ find appropriate value for $\lambda$ such that expression can be expressed as two linear factors. My Try First I thought of separately writing x terms as complete ...
0
votes
1answer
25 views

How to properly write a solutions of quadratic equation on one line?

Using this formula $$x_{1,2}={-b\pm\sqrt{b^2-4ac} \over 2a} = \space \cdots$$ what should be the correct format of the solution if I want to put both roots (x) on one line? Thanks in advance
1
vote
2answers
26 views

How many solutions are for the quadratic equation $(12+a)x^2+12ax+9a=0$

I need to give a result for each $a\in \mathbb{R}$ how many solution is there to this equation: $(12+a)x^2+12ax+9a=0$. My attempt: Check for $B^2-4AC:$ $B^2-4AC = 144a^2-4(108a+9a^2)=108a^2-432a=...
0
votes
2answers
43 views

If $ax^2+bx+c<x$, for any $x$ then is $b\ge 1$ or $b <1$ or $c=0$ or $a\le 0$?

I tried by using the following : $ax^2+bx+c<x$ $\implies (b-1)^2<4ac$ But $ax^2+bx+c<x$ also $\implies$ a parabola lying underneath a straight line $y=x$ such that the parabola faces the ...
0
votes
1answer
37 views

prove commuting quadratic functions of real numbers are equal

Suppose that $$f(x) = ax^2 +b$$ is a quadratic function, where $ (a, b) \in \mathbb R^2$ and $a \neq 0. $ If $$g(x) = cx^2 +d,$$ where $(c, d) \in \mathbb R^2$ and $c \neq 0,$ is another quadratic ...
0
votes
2answers
19 views

What is the rule to “guess” how to multiply both equations of a system so that their sum solves in “good” (perfect squares) numbers?

How did the author guess from the beginning that 1st equation must be multiplied by $3$ and the 2nd - by $17$?
0
votes
1answer
50 views

Derive the implicit cone equation from the implicit circle equation

Is it possible to derive the implicit equation of a cone $x^2+y^2-z^2=0$ from the circle equation $x^2+y^2=1$, which is the intersection between the cone and the hyperplane $\{(x,y,z)\in\Bbb R^3\,|\,z=...
2
votes
1answer
65 views

Sum of all the possible real solutions of $(x^2+4x+5)^{(x^2+4x+5)^{(x^2+4x+5)}}=2018$

Sum of all the possible real solutions of $(x^2+4x+5)^{(x^2+4x+5)^{(x^2+4x+5)}}=2018$ My try I knew that the answer is $-4$ (according to wolfram alpha) but i tried myself solving it by hand. I ...
-1
votes
1answer
67 views

Condition for both roots be infinity [closed]

For what value/s of constant 'p' for which the given quadratic have both roots as infinity. $(2p^3-13p^2+27p-18)x^2 + (2p^2-9p+9)x +2p^2-7p+6=0$ Options are :- $1) 3/2 2) 2 3) 3 4) /phi $ Since both ...
-3
votes
4answers
59 views

solving a relatively simple quadractic equation [closed]

I am young and try to solve this quadratic equation: $x^2 + 6x +5 = 0$ How can I solve it?
0
votes
1answer
25 views

Conditions when the system of symmetric quadratic equations has a solution

suppose we have a system of quadratic equations of the following form \begin{align} \boldsymbol x^T\boldsymbol A_i (\boldsymbol x + \boldsymbol b)=y_i\ ,\quad i=1,\cdots,M \end{align} where $\...
2
votes
0answers
55 views

Difficult quadratic equation

Let $$S = \sqrt{x^2+c^2+2x\left(\frac{y}{z}(c+d)-d\right)}$$ and $$B= \frac{S-x+d}{\frac{y}{z}{(c+d)}}, $$ we are supposed to find $d$ (which is independent of x) s.t.: $$ B^2y+B\left(x\left(1-\frac{...
0
votes
3answers
26 views

Solving steps for equation with exponent and addition (x^2 + x) = 2y

I have this equation: ($x^2$ + x) = 2y Which I know solves to: x = (-1 + sqrt(1 + 8y)) / 2 x = (-1 - sqrt(1 + 8y)) / 2 However, I have no idea about the steps to reach the solved equation, any ...
1
vote
2answers
106 views

If If $x^2+ax+b+1=0$ has roots which are positive integers, then $(a^2+b^2)$ can be which of the given choices?

If $x^2+ax+b+1=0$ has roots which are positive integers, then $(a^2+b^2)$ can be (A) 50 (B) 37 (C) 19 (D) 61 My approach: I first took roots $\alpha$, $\beta$ and then applied sum and ...
0
votes
0answers
67 views

Who was the first person to prove that only primes of the form $4k+1$ can evenly divide odd integers of the form $n^2+1$?

I am writing a paper and I want to cite the person(s) who proved that only primes of the form $4k+1$ can evenly divide odd integers of the form $n^2+1$. Edit: added "odd" For example, if $n=8$, then ...
2
votes
2answers
33 views

Solving quadratics writing solutions

I am confused on the notation used when writing down the solution of x and y in quadratic equations. For example in $x^2+2x-15=0$, do I write : $x=-5$ AND $x=3$ or is it $x=-5$ OR $x=3$ which ...
2
votes
3answers
66 views

How I factor $(1-p) - x + px^2$?

I came across this equation in the solution to problem 35 of "50 Challenging Problems in Probability" (page 37), which deals with a variant of the gambler's ruin problem. Specifically: A man is 1 ...
0
votes
1answer
11 views

Calculate protein concentration (x) from known absorbance

Got known absorbance ($y$) and I want to find $x$ from this formula: $$y = -4\times 10^{-7} x^2 + 0.001 x + 0.2529$$
1
vote
4answers
43 views

How do I solve $z^2+(3+4i)z-1+5i=0$

Right, I asked yesterday about the explanation for the roots of quadratic equations, now I'm trying to apply these concepts. As stated in the title, we start with: $$z^2+(3+4i)z-1+5i=0$$ If we ...
2
votes
1answer
117 views

Detailed proof that $an^2+bn+c=\Theta(n^2)$

I am reading the book "Introduction to Algorithms", 3rd Edition, by Cormen, Leiserson, Rivest and Stein, and on page 46, they write the following: "[…] consider any quadratic function $f(n)=an^2+bn+...
0
votes
1answer
74 views

Solving an interesting polynomial with degree 4? [duplicate]

So the equation is as follows: $$ 6x^2 -\ 25x \ + 12 \ +\ \frac6{x^2}\ + \frac{25}{x} = 0$$ So one thing that is immediately observable is that pairs of roots will be of the from $$x_1=-\frac{1}{...
1
vote
3answers
36 views

Help with solution to complex quadratic equation

I'm learning about solving complex quadratic equations, in the examle I'm following, we start with: $$z^2+4iz-(7+4i)=0$$ This can be rewritten as: $$(z+2i)^2-(3+4i)=0$$ The square can be rewritten ...
0
votes
0answers
26 views

Quadratic forms + and -

Be Qº, Qºº: R2 ----> R1 two quadratic forms and the following statements: (I) If Qº and Qºº are positive definite then Qº + Qºº is positive definite. (II) There are Qº and Qºº definite positive such ...
0
votes
0answers
19 views

Unable to solve quadratic equation implemented and plotted in python

I model the inverter efficiency by eff = f(P_in) which is normed by P_in_pu = P_in/P_nomresulting in ...
4
votes
6answers
143 views

For the equation $y = 4x^2 + 8x + 5$, what are the integer values of x such that y/13 is an integer?

For the equation $y = 4x^2 + 8x + 5$, what are the integer values of x such that y/13 is an integer? For example, if x = 3, $y = 4(3^2) + 8(3) + 5$ = 65 which is divisible by 13 if x = 8, y = 325 ...
1
vote
7answers
253 views

Find the minimum value of $9x^2+2y^2-8xy-6x+11$ [closed]

For any $x, y$ in real, find the minimum value of $9x^2+2y^2-8xy-6x+11$
0
votes
1answer
39 views

Solution to quadratic and cubic equation with partial root

I am having trouble understanding how to resolve quadratic and cubic equations using the method described by my university lecturer (I am really interested to know if this method has a name). My ...
0
votes
1answer
24 views

LTI System Time Constant Representation - Problem with quadratic formula

I've got a problem with factorising the denominator of an LTI system. The system is a simple boost converter and my current transfer function is $$ U_O = \frac{\lambda' \cdot U_I - I_O \cdot L \cdot ...
0
votes
2answers
37 views

Maximum vertical distance for concave function.

I need to find the maximum vertical distance between a parabola $f(x)=3-x^2$ and a line $g(x)=x+1$ on the interval of $[-2,1]$. Usually I see people doing $g(x)-f(x)$, in other words subtracting the ...
1
vote
5answers
171 views

Maximizing $ (4a-3b)^2+(5b-4c)^2+(3c-5a)^2$, such that $a^2+b^2+c^2=1 $

If $$a^2+b^2+c^2=1 $$here a,b,c are the real numbers then find the maximum value of $$ (4a-3b)^2+(5b-4c)^2+(3c-5a)^2$$ I tried to think with vectors, that is direction cosines of lines. But then the ...
1
vote
1answer
57 views

Please explain why 2x * /2 became 2/2x

I'm studying a Grade 10 maths book and was surprised to see the correct answer for this question was: (/2 - 3x)(/3 + 2x) - /5x = /6 + 2/2x - 3/3x - 6x - /5x I thought it should be: = /6 + 2x/2 - ...
1
vote
2answers
37 views

For which values of $a$ we will get two different roots?

In given the following system of equations: $$ |x-1| > 2x+2 $$ $$ x^2 + ax + a -1 = 0 $$ For which values of $a$ we will get two different roots?
0
votes
1answer
30 views

Inverse of power-2 rational function

I have a function $f(a,b) = \frac{ab}{(\frac{a+b}{2})^2}$, and (to me) it has some cool properties (e.g $f(a,b) = f(b,a)$, $f(x,0) = 0$, $f(x, x) = 1$, $0 \leq f \leq 1$, etc.). Now I wanted to know ...
0
votes
0answers
20 views

Solving quadratic equations in $\mathbb F_{2^h}$ using the sum and product rule.

In the finite field $\mathbb F_{2^h}$ with primitive element $\alpha$ consider the quadratic eqution: $$\alpha^ax^2+\alpha^b x+\alpha^c =0$$ In class we saw we could solve it using the subsition $y=\...
-1
votes
5answers
57 views

Let $a \in \mathbb R$. Prove that ($x^2 + ax + a > 0$ for every $x \in \mathbb R)$ iff $(0< a<4)$ [closed]

List item This is a homework question for my university math proofs course. I am asked to prove the above bi conditional statement. The text book gives the hint that completing the square of the ...
0
votes
1answer
48 views

Have any idea of a way to solve this equation? I don't have any software to solve this.

So here is my problem. I have no idea about how to solve this equation and i am not even working on it. It seems gigantic. I've been Googling, but can't narrow it down. I have tried to use auxiliary ...
1
vote
2answers
85 views

The number of prime pairs of $x^2-2y^2=1$

How to find the number of pairs of positive integers $(x,y)$ where $x$ and $y$ are prime numbers and $x^2-2y^2=1$? I am not getting any clue here.
9
votes
3answers
1k views

Let k be an integer. Disprove: “The equation $x^2 − x − k = 0$ has no integer solution if and only if $k$ is odd.”

My problem is I keep ending up proving the statement true, instead of disproving it. I was getting it mixed up in my mind so I broke it down into very explicit steps but now I'm wondering if I'm ...
2
votes
2answers
142 views

Quiz Question: Solve for $x$: $9^{2x+1} - 28 (3^{x}) +3 = 0$? [closed]

I know this is probably quite basic, but I've got my school mathematics quiz tomorrow and I've honestly drawn a blank with this question: Find the sum of the roots of the equation: $9^{2x+1}-28(3^x)+...
0
votes
2answers
184 views

Solving a quadratic equation using the “splitting the middle term” method.

Use splitting the middle term method to solve the below equation. Is there a limitation to this method? $$5b^2-16b+4=0$$
3
votes
4answers
65 views

What is the number of elements in the solution set of $(x^2-4)^2\cdot(x^2-6x-7)=0$?

$(x^2-4)^2\cdot(x^2-6x-7)=0$ $S.S.=\{x_1,x_2,...,x_n\}$ $\Rightarrow n=?$ Answer is given as $4$. I think it should be $6$ because of multiplicity of the roots. I debated this problem with ...
0
votes
1answer
37 views

Quadratic eigenvalue problem (QEP)

$Q(\lambda)x=0$ and $Q(\lambda) = \lambda^2 M+\lambda C +K$ are defined in this PDF file The matrices $M$, $C$, $K$ are $n\times n $ matrices. The thesis said that when $M$, $C$, $K$ are real or ...
-1
votes
1answer
44 views

If the roots of the equation $6x^2-7x+K=0$ are rational, then is equal to–

If the roots of the equation $6x^2-7x+K=0$ are rational, then is equal to: $1)$ $-2$ $2)$ $-1,-2$ $3)$ $-2$ $4)$ $1,2$
-3
votes
3answers
55 views

Solve quadratic equation with two variables

I can't find a solution for this equation $$\frac{(x-2016)(y-2017)}{(x-2016)^2 + (y-2017)^2}= -\frac{1}{2}$$ Any help?