Tagged Questions

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where am I going wrong with solving this equation?

solve $z^2=2e^{5{\pi}i/6}$. Well, clearly $z={\sqrt{2}}e^{5{\pi}i/12}$ is a root so its' conjugate $z={\sqrt{2}}e^{-5{\pi}i/12}$ is the other root. But I can also argue ...
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Can you, given a large number N, find a, b, c such that ax^2 + bx + c = 0 has at least N roots? All of this is in any mod you choose.
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Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
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How to solve the following? $x^3+1=2{(2x-1)}^{1/3}$.

Find all the real solutions of $$x^3+1=2{(2x-1)}^{1/3}$$ I tried to cube both sides but got messed up with a nine degree equation! Please help. Thanks in advance!
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Request for help with a quadratic polynomial question.

If the rooots of the equation $x^2+bx+c=0$ are real , show that the roots of the equation $x^2 +bx+c(x+a)(2x+b)$ are again real for every real number a. I assumed the discriminant of the first ...
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Fitting a quadratic polynomial to two points such that it is always concave downward

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, I'd like to construct a quadratic polynomial of the form $y = a_2x^2 + a_1x + a_0$ such that it intersects both points and is concave downward (i.e., ...
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What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
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Evaluate $a+b+c+d$

If $a$, $b$, $c$, and $d$ are distinct integers such that $$(x-a)(x-b)(x-c)(x-d)=4$$ has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$? I tried to analyze graphically by ...
Find the value of $\left | b-c \right |$
Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
Problem : Let f be a continuous function defined on [-2009,2009] such that f(x) is irrational for each $x \in [-2009,2009]$ and $f(0) =2+\sqrt{3}+\sqrt{5}$ Prove that the equation $f(2009)x^2 +2f(0)x ... 1answer 76 views Factors of integers of the form$k^2-k+1$Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form$k^2-k+1$? Is there some way to compute ... 0answers 106 views Consider the quadratic equation$ax^2-bx+c=0, a,b,c \in N. $If the given equation has two distinct real root… Problem : Consider the quadratic equation$ax^2-bx+c=0, a,b,c \in N. $If the given equation has two distinct real roots belonging to the interval$(1,2) $then the minimum possible values of a is ... 1answer 69 views A Cubic Equation$2x^3+ax^2+bx+4=0$,$(a,b \in R^+)$has three real roots. Then : A.$a\geqslant 4.2^{\frac 1 3}$B.$a\geqslant 1.2^{\frac 1 3}$C.$a\geqslant 6.2^{\frac 1 3}$D.$a\geqslant 2.2^{\frac 1 3}$... 1answer 59 views How to interpret coefficients in polynomial regression? I am working on my thesis (study) about poverty incidence rate and its socio-economic factors using second-order polynomial regression without interaction. The final model in my study is ... 1answer 21 views Verification of solutions to some polynomial prob/$\boxed{\text{Problem 1}}$Find the other solution of the equation$(1+\sqrt3)x^2-(5-\sqrt3)x+6-6\sqrt3=0$given that$2$is a solution Ma solution:$x_1\cdot x_2=\tfrac ca$therefore letting ... 2answers 37 views prove for p(x) which is a quadratic polynomial$p(x)$is a quadratic polynomial . Prove that any given number for$a$with one exception , we can find a number$b$such that$p(a)=p(b)$and$a$is not equal to$b$. 3answers 55 views A question on quadratic equations.. Given below in the picture. PLease also tell how u got to the answer as I want to know the way to solve further questions 1answer 130 views Finding two unknowns in two quadratic polynomials with only knowing the divisors There are two quadratic polynomials (dividends). These two polynomials are divided by two different linear polynomials like$x+1$(divisors). The remainders are known, but the quotients are unknown. ... 2answers 178 views Solving system of multivariable 2nd-degree polynomials How would you go about solving a problem such as: \begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix} where$(x,y)\in\mathbb{C}^{2}$. More generally, how would ... 1answer 70 views Solving a quadratic made from the sum of monomial denominators. [closed] Solve the following equation. Separate your answers with commas. Repeated roots should only be entered once. $$\frac{1}{x-5} + \frac{1}{x-6} = \frac{11}{30}$$ Any ideas on how to start out? 2answers 846 views coefficients of quadratic function? In a quadratic function: coefficient$a$controls the speed of increase/decrease from the vertex. coefficient$b$controls the downward slope as the function crosses the y-axis. I don't really ... 2answers 1k views Finding the discriminant and roots of a polynomial How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where$f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ and here$\Delta$is the ... 2answers 59 views How do you know which substitutions to make to cancel out a term? I am doing problem B45 from Ivan Niven's "Maxima and Minima Without Calculus" which says: "Consider the quadratic polynomial$f(x, y)=ax^2+2bxy+cy^2+dx+cy+k$, where the coefficients are real ... 1answer 106 views Quadratic Polynomial Question - Solving for a coefficient using the discriminant This question has been troubling me: A parabola whose equation is of the form$y = Bx^2$(where B is a constant) has the line$20x - y + 20 = 0$as a tangent. Find$B$. The explanation says, ... 3answers 151 views Quadratic Equation find the value of$\lambda$when other roots are given in restriction Problem : If$\lambda$be an integer and$\alpha, \beta$be the roots of$4x^2-16x+\lambda$=0 such that$ 1 < \alpha <2$and$2 < \beta <3$, then find the possible values of$\lambda$... 1answer 216 views Finding the rational values of constant for which these constants are roots of equation Problem : Determine all rational values for which$a,b,c$are the roots of$x^3+ax^2+bx+c=0$Solution : Sum of the roots$a+b+c = -a$........(i) ( Since , as per question$a,b,c$are roots of ... 3answers 156 views Is it possible to find out$x^2$parabola and function from 3 given points? I am programming a ball falling down from a cliff and bouncing back. The physics can be ignored and I want to use a simple$y = ax^2$parabola to draw the falling ball. I have given two points, the ... 1answer 97 views Finding a polynomial of degree$n$when value of$f(k)$is equal to some value Problem : If$f(x)$is a polynomial of degree$n$and if$f(k) = \frac{k}{k+1}$where$k =0,1,2,\ldots,n$, find$f(x)$. Can we go like this : Let the polynomial be ... 1answer 85 views what if geometric sequence + geometric sequence I wrote a program that basicly can find the formula of the sequence that created with any-degree equation. For example if you give my program that sequence: [1926, 2811, 833240, 28778265, 398155842, ... 1answer 605 views Getting square root of negative in completing the square problem I try to solve the equation$f(x) = 7x - 11 - 2x^2 = 0$for$x$, but run into troubles. I've gone through it over and over again as well as similar problems, but can't find what I'm doing wrong. ... 2answers 49 views How do I transform the equation based on the condition? If$q$and$w$are the roots of the equation $$2x^2-px+7=0$$ Then$q/w$is a root of ? P.s:- It is an another question of How do I transform the equation based on this condition? 1answer 44 views How do I transform the equation based on this condition? If a and b are the roots of the equation $$2x^2-px+7=0$$ Then a-b is a root of ? 2answers 128 views How do I proceed with these quadratic equations? The question is $$ax^2 + bx + c=0$$ and $$cx^2+bx+a=0$$ have a common root, if$b≠ a+c$, then what is $$a^3+b^3+c^3$$ 5answers 228 views Proving Quadratic Formula purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving$ax^2 + bx + c = 0$." section. How does$\dfrac{b}{2a}$become ... 4answers 4k views Is it possible for a quadratic equation to have one rational root and one irrational root? Is it possible for a quadratic equation to have one rational root and one irrational root? Yes, a pretty straightforward question. Is it possible? 4answers 762 views A quadratic equation$ax^2+bx+c=0$has equal roots at$a=2c$. How could we find the sum of reciprocals of the roots of this equation? A quadratic equation$ax^2+bx+c=0$has equal roots at$a=2c$. How could we find the sum of reciprocals of the roots of this equation? I need some hints for solving this problem. 16answers 10k views Why can ALL quadratic equations be solved by the quadratic formula? In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ... 6answers 263 views Factoring Quadratics Is there a method to find which numbers to use when simplifying quadratics? For example$x^2 + 5x + 6$is easy enough to find, but what if I have bigger numbers, or I have this quadratic expression: ... 5answers 298 views How to “Re-write completing the square”:$x^2+x+1\$
The exercise asks to "Re-write completing the square": $$x^2+x+1$$ The answer is: $$(x+\frac{1}{2})^2+\frac{3}{4}$$ I don't even understand what it means with "Re-write completing the square".. ...