Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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

0
votes
1answer
29 views

Determining whether a quadratic has a maximum or minimum

So I've learnt that quadratics equations with a positive coefficient on the squared term have a minimum and a maximum if the coefficient is negative. But if we rearrange the quadratic and change the ...
4
votes
2answers
113 views

Is it true that $(a^2-ab+b^2)(c^2-cd+d^2)=h^2-hk+k^2$ for some coprime $h$ and $k$?

Let us consider two numbers of the form $a^2 - ab + b^2$ and $c^2 - cd + d^2$ which are not both divisible by $3$ and such that $(a, b) = 1$ and $(c,d) = 1$. Running some computations it seems that ...
-3
votes
1answer
44 views

Simplification of identity with square roots [closed]

$\dfrac{\sqrt{x + 1}}{2x + 1} + \dfrac{\sqrt{2x + 1}}{x + 1} = 1 \tag 1$ How can I find the value of $x$ in this question?
0
votes
2answers
58 views

Change the vertex of a parabola while ensuring it still passes through a particular point

I have a parabola defined by the quadratic equation $y = -(x + 0)(x - endPoint)$, which also passes through a particular point $(a, b)$. I would like to know how to alter the equation so that I can ...
0
votes
1answer
29 views

Finding the steady states of a quadratic ODE

How would I go about finding the steady states I know I need to set $\frac{dx}{dt}=0$ but then I'm struggling with the next step.
2
votes
3answers
45 views

If $\alpha$, $\beta$ are the roots of the equation $ax^2 + 3x + 2 = 0,\; a<0$

Show that if $\alpha$, $\beta$ are the roots of the equation $ax^2 + 3x + 2 = 0,\; a<0$, then $$\dfrac{(\alpha^2)}{(\beta)}+\dfrac{(\beta^2)}{(\alpha)}> 0$$ I could only figure out two things ...
0
votes
1answer
41 views

Solving simple quadratic - Wolfram Alpha confusion?

I have the following quadratic $$(2\sqrt 2 - 2)x^2 + \sqrt8 x + (1+\sqrt 2)=0$$ Now the discriminant of this is $0$, so it has one real repeated root. A plot on Desmos confirms this. However, ...
1
vote
3answers
50 views

If the equation $\sin^2x-a\sin x+b=0$ has only one solution in $(0,\pi)$, then what is the range of $b$?

If the equation $\sin^2(x)-a\sin(x)+b=0$ has only one solution in $(0,\pi)$, then what is the range of $b$? What I try: $$\displaystyle \sin x=\frac{a\pm \sqrt{a^2-4b}}{2}\in\bigg(0,1\bigg]$$ for ...
0
votes
1answer
240 views

How to find quadratic regressions by hand

Currently I am working on an assignment for which I have to calculate the quadratic regression and linear regression (I know how to do this one) of some data points by hand. Nonetheless, I do not know ...
1
vote
1answer
79 views

To what sets must $a,b,c$ belong?

I just thought of kind of a cool number theory/algebra problem. Given that $$\sqrt{b^2-4ac}\in\Bbb N$$ To which sets must $a,b,c$ belong? It is obvious that $$b^2-4ac\in\left\{x^2|x\in\Bbb N\...
1
vote
1answer
26 views

Solve one side of irregular quadrilateral with known area (formula help)

My question is about determining the formula for a missing length of a known area irregular quadrilateral. Its a fairly easy one, but i'm about 40 years outside of my math lessons! It's a land area ...
0
votes
4answers
114 views

Proving $1 +\sqrt[3]{5-\sqrt2}$ is rational via the rational roots theorem [closed]

Find a polynomial that has $x = 1 +\sqrt[3]{5-\sqrt2}$ as a root, then use rational roots theorem to show that $x$ must be rational. I came up with the polynomial $100,000,000,000x^2 = 640393215809$ ...
0
votes
2answers
45 views

Arc length of quadratic curve

I would like to find the arc length of a curve from $a\le t\le b$, the curve is $t^2A+tB+C$ $$arcLength=\int_{a}^{b}\sqrt{(2At+B)^2+1}\,dt$$ I am having trouble getting rid of $t$ (the variable) (...
0
votes
2answers
24 views

How to solve two 2 variable quadratics using system of equations?

Given $\left\{\begin{array}{rcrcr} {\displaystyle\left(x + 2\right)^{2}} & {\displaystyle +} & {\displaystyle\left(y - 2\right)^{2}} & {\displaystyle = } & {\displaystyle 9} \\[1mm] {\...
0
votes
1answer
56 views

Solve the equation $R^{2}\frac{\arccos(\frac{h}{R})+\frac{h}{R}}{2} = \pi r^{2}$ for $R$

I have the following equation and I want to solve for $R$. $$R^{2} \left( \frac{\arccos \left(\frac{h}{R}\right) + \frac{h}{R}}{2} \right) = \pi r^{2}$$
3
votes
1answer
63 views

Common root of cubic and quadratic equation

If equations $ax^3+2bx^2+3cx+4d=0$ and $ax^2+bx+c=0$ have a non zero common root, prove that $(c^2-2bd)(b^2-ac) \geq 0$. I know the condition of common root of two quadratic equations but I have no ...
0
votes
0answers
34 views

The quadratic function problem

$H(p,q)$ and $I(r,s)$ are two points on $f(x)=x^2-6x+11$. If $p+q=6$, what is the relationship between $q$ and $s$?
4
votes
0answers
55 views

Help me see the connection between exponential functions and quadratic curves

We know that the degree 2 equations $x^2 + y^2 =1$ and $x^2 - y^2 =1$ can be parametrized by exponential functions. How come exponential functions show up in this seemingly unrelated area? I think it ...
1
vote
1answer
86 views

Why is $b^2-4ac<0$ if a linear line and a curve do not meet?

For example there's a curve $y=X^2-4X+7$ and a line $L: Y=mX-2$. Both never intersect at any given point. If we were to find the set of values of $m$ for which $L$ does not meet the curve, the ...
8
votes
3answers
72 views

Find $\max\{y-x\}$

If $x+y+z=3, $ and $x^2+y^2+z^2=9$ , find $\max\{y-x\}$. I tried to do this geometrically, $x+y+z=3$ is a plane in $\Bbb{R}^3$ and $x^2+y^2+z^2=9$ is a ball with radius 3 and center of origin . So ...
1
vote
1answer
30 views

Using Nature of Roots to find range of values

Find the range of values of $k$ for which $3x^2-4(k-x)+2$ is always positive for all real values of $x$. I've tried simplifying, until I got to: $3x^2-4k+4x+2$. Since it must always be positive, the ...
2
votes
2answers
54 views

Decide for which $x,y,z$ the following equation system is met: ${1+x+y=xy}$ $2+y+z=yz$ and $5+z+x=zx$

I need to decide for which $x,y,z$ the following equation system is met: $$ 1+x+y=xy $$ $$ 2+y+z=yz $$ $$5+z+x=zx$$ I can see that $x=0, y=0, z=0$ is not a solution. I tried to divide ...
0
votes
0answers
17 views

create quadratic curve $R$ distance from another quadratic curve

I have a curve suported by three points, I would like to create an outside rail that is always distance $R$ from the other curve so far I have something like this Above is my attempt, but under ...
0
votes
2answers
18 views

Isolating a variable under square root

Given this equation: $T=\sqrt{(ugx)^2+(T_0)^2}$ You're asked to isolate $x$. My process was: $T=ugx + T_0$ (the square root cancelled the exponents) $T-T_0=ugx$ $x=\frac{T-T_0}{ug}$ But that was ...
2
votes
2answers
70 views

Condition for $x^4-18x^2+4dx+9=0$ has four real roots

Prove that if $x^4-18x^2+4dx+9=0$ has four real roots, Then $d^4 \le 1728$ My try: obviously the equation should have two unequal negative roots and two repeated positive roots OR vice versa. ...
1
vote
3answers
41 views

Write $f(x) =x^3+x^2-3x-3$ as a product of a linear factor and a quadratic factor [closed]

I know that the linear factor is $(x+1) (x^2-3),$ but how would I find the quadratic factor?
2
votes
2answers
46 views

How to find the number of roots for a constant in a quadratic equation when the independent term is unknown?

I'm going in circles with this question. As I don't know how to deal with the fact that the independent term is unknown. Typically when solving a quadratic equation you know the terms or they can be ...
0
votes
2answers
63 views

Equation of parabola that passes through two points and vertex has coordinates ($x_v$, $0$)

I can't solve the last exercises in a worksheet of Pre-Calculus problems. It says: Quadratic function $f(x)=ax^2+bx+c$ determines a parabola that passes through points $(0, 2)$ and $(4, 2)$, and its ...
0
votes
1answer
60 views

Where does this answer come from? $2x^2+4x+c-1=0$

Note: I could do a simple algebra solution but the question wants me to answer this question using something that relates to parabola function. That question is "if $2x^2+4x+c-1=0$ what is c?" the ...
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
76 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
69 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
47 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
66 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
24 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
104 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
66 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 ...