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
0answers
33 views

How to factor $-x^2 + x -10$?

This question has been killing me for hours. None of the factors of $10$ (because $(-10)(-1) = 10$) add up to $1$. So how do you do this question?
0
votes
0answers
21 views

Quadratic equation, absolute value of roots strictly superior to 1 conditions

Let's consider the equation: \begin{align} 1 - \phi_1 z - \phi_2 z^2 = 0 \end{align} We want to find the conditions on $\phi_1, \phi_2$ for the roots to have an absolute value strictly superior to 1. ...
0
votes
2answers
25 views

Solving quadratic equation for inverse variable

I'm reading through some lecture notes and they show a quadratic equation, which I will just write in the usual way as $$ax^2+bx+c=0$$ The notes say that, even though that equation can be solved in ...
0
votes
1answer
18 views

Max possible area, of a rectangle shape where one side is a half circle. circumference of 100m

A picture of the shape! I recently took a maths test where one of the questions was just unsolvable for me. I'm going to try to make it as clear as possible, to not create confusion. The question ...
5
votes
4answers
685 views

What is the Difference Between Formulating the Answer via Quadratic Formula and Factoring?

I'm quite eager to learn what is the difference between factoring quadratics (the $(x + a)(x + b)$ method), and using the typical formula (where $x = (-b \pm \sqrt{b^2 - 4ac})/2a$), and in what ...
1
vote
1answer
30 views

Solving a System of Quadratic Equations for Sound Triangulation

I am currently attempting to solve a system of quadratic (and linear) systems that I have run into while trying to triangulate sound. My hypothetical setup includes 3 sensors on a perfectly ...
-1
votes
2answers
24 views

What is the maximum value of $x^2+4xy-y^2$ for all $(x,y)$ satisfying $x^2+y^2 = 1$? [on hold]

Does the trick have something to do with the equation of a circle?
0
votes
3answers
62 views

Solve for $x$ in $x^2-5x+2\sqrt{x^2-5x+3}= 12$

Solve for $x$ in $x^2-5x+2\sqrt{x^2-5x+3}= 12$ I've tried moving root term to one side and squaring both sides to get a $4$th-degree polynomial and find the roots that way. Is there any easier way of ...
0
votes
0answers
19 views

Calculate quadratic function from given points in 2 dimensions

The quadratic function can be defined as $z = a+b*x+c*y+d*x^2 +e*x*y+f*y^2$ But how to find the 6 coefficients from given truples $(x_i,y_i,z_i)$? I hope that there is a not too difficult solution. ...
2
votes
2answers
22 views

Prove $(X \theta - \vec{y})^T (X \theta - \vec{y}) = \theta^T X^T X \theta - \theta^T X^T \vec{y} - \vec{y}^T X \theta + \vec{y}^T \vec{y}$

I'm studying Machine Learning Stanford's CS229 course and in the lecture note, page number 11, I'm not getting how does step 2 arrive from step 1 above? Prof. Andrew Ng says that it is the expansion ...
1
vote
3answers
28 views

Prove that for $a,p,q \in \Bbb R$ the solutions of: $\frac{1}{x-p} + \frac{1}{x-q} = \frac {1}{a^2}$ are real numbers.

Prove that for $a,p,q \in \Bbb R$ the solutions of: $$\frac{1}{x-p} + \frac{1}{x-q} = \frac {1}{a^2}$$ are real numbers. I tried manipulating the expression, getting rid of the denominators, but i ...
1
vote
2answers
39 views

Find values of $a$ such that $x^2+ax+a^2+6a \lt 0$ $\forall$ $x \in (1,2)$

Find values of $a$ such that $x^2+ax+a^2+6a \lt 0$ $\forall$ $x \in (1,2)$ My try: Since $y=x^2+ax+a^2+6a$ is an open upward Parabola, the roots $\alpha,\beta$ should be distinct and satisfy $1 \lt \...
0
votes
2answers
20 views

Quadratic equation solving

How to prove that the value of x from $x^{2}+y^{2}=a^{2}$ and $y=mx+c$ are equal when $c^{2}=a^{2}(m^2+1)$? I tried to equate the x from both variables but can't get it
0
votes
0answers
11 views

quadratic bezier to parabola matrix equation

I'm trying to follow the matrix equation solution presented here: Convert quadratic bezier curve to parabola by @robjohn I'm assuming his solution can be used for any quadratic coordinates. My goal ...
2
votes
3answers
39 views

Graphical method of solving quadratic and cubic equations

I once came across a method for solving quadratic and cubic equations using a graphical method as shown below (where the lengths of the line segments are equal to the coefficients of the equation). ...
0
votes
2answers
60 views

Find the relation between $m$ and $n$ such that the following equation has four roots. [closed]

Find the relation between $m$ and $n$ such that the following equation has four roots with $m > 0$. $$x^2 + \left(\dfrac{mx}{m + x}\right)^2 = n$$ Well, I know what the answer is. I just want to ...
1
vote
2answers
45 views

How do we compute $\sqrt[3]{x_1} +\sqrt[3]{x_2} $ using the fact that $x_1 + x_2 = 4 , x_1x_2 = -1$? [closed]

Given quadratic equation $$x^2 -4x-1 = 0$$ How do we compute $\sqrt[3]{x_1} +\sqrt[3]{x_2} $ using the fact that $x_1 + x_2 = 4 , x_1x_2 = -1$? Regards
2
votes
1answer
40 views

why do we factor the roots of an equation as $x-x_1$ and not $x+x_2$

why do we factor the roots of an equation as $x-x_1$ and not $x+x_2$. In the quadratic formula b= the sum of the roots multiplied by the leading coefficient and -.
0
votes
1answer
17 views

Express m in terms of k given a quadratic function [closed]

The quadratic function $h(x) = -2x^2 + 4x - m$ can be expressed in the form of $h(x) = 7 - 6k - (x - 2k)^2$ where $k$ and $m$ are constants. Express $m$ in terms of $k$.
4
votes
1answer
129 views

(Calculus) Derivative Thinking Question

Recently, my Calculus and Vectors (Grade 12) teacher gave our class a thinking question/assignment to work on over the march break, and after working on for some time, I've become stuck on it. The ...
3
votes
2answers
91 views

Why doesn't a parabola have two tangents at its vertex?

Perhaps my definition of 'tangent' is the problem but in school the tangent is always defined as a line that intersects with a curve at only one point. According to this definition the equation $y = x^...
3
votes
3answers
78 views

Number of real solutions of the equation $6x^2 -77[x] + 147 =0 $.

How many real solutions of $$6x^2 -77[x] +147=0$$ are there, where $[x]$ is the integral part of $x$? The answer says 4 solutions but I got none. As: $6x^2 + 147 = 77[x]$ LHS= integer Therefore, ...
2
votes
3answers
52 views

Some questions on the intersection of three cones.

I have three cones in $\mathbb{R}^3$, explicitly defined by the equations: $$ (x-\alpha_x)^2+(y-\alpha_y)^2=(z-r_1)^2 \,, \\ (x-\beta_x)^2+(y-\beta_y)^2=(z-r_2)^2 \,, \\ (x-\gamma_x)^2+(y-\gamma_y)^2=(...
1
vote
1answer
44 views

Determine the equation of the largest circle inscribed in an ellipse

I have an equation of an ellipse (we know that this is an ellipse): $$7x^2 -4xy + 4y^2-6x - 12y = 9. $$ How do we determine, using linear algebra, the equation of the largest circle inscribed in ...
2
votes
1answer
15 views

Proof request: All values a for which a quadratic-linear system has exactly one point of intersection

Let $f$ and $g$ be the functions defined by $f(x)=x+2$ and $g(x)=(x^2)−a$, where $a$ is a positive constant. What are all values of $a$ for which the graphs of $f$ and $g$ have exactly one point of ...
0
votes
1answer
26 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 ...
-1
votes
0answers
23 views

Improper Factoring Method?

I have the equation $3x^2-6x-24=0$ and am curious if I coose to solve in the following way: $3x^2-6x=24 \rightarrow x^2-2x=8 \rightarrow x(x-2)=8$ So I can conclude that the only numbers ...
4
votes
2answers
106 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
54 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
28 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
35 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
38 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
45 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
29 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
76 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\...
0
votes
1answer
20 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
107 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
31 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
23 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
45 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 $R$. $$R^{2} \left( \frac{\arccos \left(\frac{h}{R}\right) + \frac{h}{R}}{2} \right) = \pi r^{2}$$
2
votes
1answer
41 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
33 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
50 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
70 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 ...
7
votes
3answers
61 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 ...
0
votes
1answer
29 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 ...
1
vote
2answers
49 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
15 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
16 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 ...