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

3,663 questions
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Relationship between Discriminant and coefficients of a Quadratic function

To answer question (b), the problem requires me to use the discriminant. However, I still do not understand how would I realize that I would have to make use of it. Can someone please explain the ...
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Finding minimum value of relation with quadratic coefficient

Let $f(x)= ax^2 +bx+c$, such that $f(x)\geq0$. Find minimum value of $\frac{f(1)}{f(0)-f(-1)}.$ My attempt: denominator should be maximum so $f(-1)=0$ as $f(x)$ is always greater than or equal to ...
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What causes the extraneous root when intersecting parabola $y^2=4ax$ with circle $x^2+y^2=9a^2/4$?

If I solve the parabola: $y^2=4ax$ and the circle: $x^2+y^2=\frac{9a^2}{4}$ I get a quadratic in $x$; ie: $$4x^2+16ax-9a^2=0$$ which has roots $x=\frac{a}{2},-\frac{9a}{2}$. But if we see the graphs ...
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Number of different quadratic functions mod 12.

How many different quadratic functions are there of form: $x \mapsto ax^2+bx+c \pmod{12}$ All I could come up with is an upper bound of $11*12*12$. This is a puzzle from a YouTube video by ...
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If $x^2 +135= x^2 −12$ , then what is the value of $x$? By considering ring. [closed]

How to find the value of $x$ in this types of equations, as I know no real solutions exist, but how to find imaginary solutions
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Solving $x^2-2x-3<0$

If i have to solve $x^2-2x-3<0$ I would do $$x+1 < 0, \quad x-3<0$$ and end up getting $x<-1$ and $x<3$. Why is it wrong to use the same inequality sign? Shouldn’t both of the ...
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Solving in terms of z , three variable two equation system

Solve in terms of $z$ $$\begin{cases} 4z&= x + 2y \\ 3z^2&=\frac{1}{2}x^2 + y^2 \\ \end{cases}$$ Solution: $x = 2z/3$ and $y = 5z/3$. I don't understand how they got to the solution with ...
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Solving differential equation by factorization

I'm trying to solve the following differential equation: $(x^{2} -1) y'' - (4x^{2} -3x-5) y' + (4x^{2} -6x -5) y = e^{2x}$ By the method of factorizing operators, so I rewrite the differential ...
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I have the quadratic equation $x^2-5x+7-i=0$, I am not sure how to solve? I was going to use the quadratic formula but I wasn't sure as the quadratic equation seems quite easy and working out $\sqrt{-... 4answers 54 views Finding values for K such that the roots of the quadratic are strictly imaginary:$x^2+\left(K^2+3K-7\right)x+Kx^2+\left(K^2+3K-7\right)x+K$I'd like to know the general approach needed to find out how to find solutions for K when I want the roots of this equation to have a specific property, such as ... 2answers 54 views Find the values of$a,b \in \Bbb R$(if exists) such that$-5 \le \frac{x^2+ax+b}{x^2+2x+3} \le 4$for all$x \in \Bbb R$Find the values of$a,b \in \Bbb R$(if exists) such that $$-5 \le \frac{x^2+ax+b}{x^2+2x+3} \le 4$$ for all$x \in \Bbb R$My try: I noticed that$x^2+2x+3 > 0$, so i can divide the inequality ... 1answer 41 views Find (a,b,c) if$ax^2-bx+c=0$have roots lying in$(0,1)$where$a,b,c\in\mathbb{Z_+}$[duplicate] Let the equation$ax^2-bx+c=0$have distinct real roots both lying in the open interval$(0,1)$where$a,b,c$are given to be positive integers. Then the value of the ordered triplet$(a,b,c)$can be ... 1answer 44 views Derivative of a multivariate quadratic function Let's define function$f : \mathbb{R}^n \to \mathbb{R}$as $$f(x) = {1\over2}x'Ax + b'x$$ where matrix$A \in \mathbb{R}^{n\times n}$and vector$b\in \mathbb{R}^n$are given. Function$f$is ... 3answers 39 views In solving Real World Quadratic Equations, why does 𝑐 change value, and when do i have to change value? [closed] When solving for the equation I entered 32, not -32. When I watched the video paired with this course it did not explain in what cases to do this. All help is appreciated :D 1answer 64 views If a,b,c are positive rational numbers such that a>b>c then tell which of the following statement are correct following quadratic equation I am solving following question based on quadratic equation If$a,b,c$are positive rational numbers such that$a>b>c$and the quadratic equation$(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$has a root ... 4answers 131 views minimum of$a^2+4b^2+c^2$given$2a+b+3c=20$If$a,b,c\in\mathbb{R}$and$2a+b+3c=20.$Then minimum value of$a^2+4b^2+c^2$is what i try Cauchy schwarz inequality $$(a^2+(2b)^2+c^2)(2^2+\frac{1}{2^2}+3^2)\geq (2a+b+3c)^2$$ How do i solve ... 2answers 67 views Is value of$\alpha$defined? Consider three quadratic functions: $$P_1(x)=ax^2-bx-c$$ $$P_2(x)=bx^2-cx-a$$ $$P_3(x)=cx^2-ax-b$$ Where$a,b,c \in \mathbb{R}\backslash \left\{0\right\}$If there exists real number$\alpha$such ... 1answer 49 views What is the range of$(f+g)(x)$where$f(x)=x^2+4x-3$and$g(x)=3x^2-8x+9$? What is the range of$(f+g)(x)$? I plugged in the domain values to get the range for each of the equations and then I would have summed them up. But I'm getting erroneous results. Because each ... 0answers 26 views Finding how many intersections two parabolas have within a certain domain I need to find out if$y=x^2-\sqrt{200}x+50$and$y=a(x-\sqrt8)^2$how for what values of$a$they will have only one point of intersection within the domain$[\sqrt8,\infty)$? 1answer 98 views On monotonic quadratic least squares Quadratic least squares can be used to fit a quadratic curve to$3$or more points, such that the resulting curve is the quadratic curve that has the least squared distance of the data points to the ... 4answers 53 views how to memorize the sum and product of roots for an$n^{th}$degree equation For my exams I need to know the following equations by heart: for a polynomial equation:$a_nx+a_{n-1}x^{n-1}+...+a_1x+a_0,the sum and product of the roots are given by \textrm{Sum}=-\frac{a_{n-... 2answers 79 views Analytical solution to polynomial system I have a polynomial system with three equations and three unknowns i wish to solve analytically. I can obtain a numerical solution easily but for my project i need to find a analytical solution. The ... 1answer 52 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? 0answers 23 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. ... 2answers 29 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 asax^2+bx+c=0$$The notes say that, even though that equation can be solved in ... 1answer 23 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 ... 4answers 723 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 ... 2answers 92 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 ... 2answers 29 views What is the maximum value of x^2+4xy-y^2 for all (x,y) satisfying x^2+y^2 = 1? [closed] Does the trick have something to do with the equation of a circle? 3answers 67 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 4th-degree polynomial and find the roots that way. Is there any easier way of ... 0answers 23 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. ... 2answers 32 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 ... 3answers 31 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}$are real numbers. I tried manipulating the expression, getting rid of the denominators, but i ... 2answers 42 views Find values of$a$such that$x^2+ax+a^2+6a \lt 0\forallx \in (1,2)$Find values of$a$such that$x^2+ax+a^2+6a \lt 0\forallx \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 \...
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