Roots of equations Definition and 12 Threads

The Solution of the Quadratic Equation By Differentiation Method

On simplifying the given equation we get, x^2-x-1=0 and using the quadratic formula we get x=(1+√5)/2 and x=(1-√5)/2 Now, as the formula suggests, there are two possible values for x which satisfies the given equation. But now, if we follow a process in any general calculator by entering...

Given : The quadratic equation ##x^2+px+q = 0## with coefficients ##p,q \in \mathbb{Z}##, that is positive or negative integers. Also the roots of the equation ##\alpha, \beta \in \mathbb{Q}##, that is they are rational numbers. To prove that ##\boxed{\alpha,\beta \in \mathbb{Z}}##, i.e. the...

It is given that ##x_1, x_2\; \text{and}\; x_3## are roots of the equation ##ax^2+bx+c=0##, which are pairwise distinct. If indeed they are roots, we should have ##ax_1^2+bx_1+c= 0 = ax_2^2+bx_2+c= 0 = ax_3^2+bx_3+c= 0##. On subtracting the first two, we obtain ##a(x_1^2-x_2^2)+b(x_1-x_2) =...

Mentor note: Thread moved to Diff. Equations subforum Hello, few days ago I had a calculus test in which I had to find the general solution for the next differential equation: (D^8 - 2D^4 + D)y = 0. "D" stands for the differential "Dy/Dx". However I could only find 2 of the roots on the...

For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?

Homework Statement Let ##a,b,c## be positive integers and consider all the quadratic equations of the form ##ax^2-bx+c=0## which have two distinct real roots in ##(0,1)##. Find the least positive integers ##a## and ##b## for which such a quadratic equation exist. Homework EquationsThe Attempt...

Is there a good way to relate the symmetries of the graphs of polynomials to the roots of equations? There's lots of material on the web about teaching students how to determine if the graph of a function has a symmetry of some sort, but, aside from the task of drawing the graph, I don't find...

$a=1,2,3,4,5,------2011$, the roots of the equations $x^2-2x-a^2-a=0,$ are : $(\alpha_1,\beta_1),(\alpha_2,\beta_2),----------,(\alpha_{2011},\beta_{2011})$ respectively please find : $\sum_{n=1}^{2011}(\dfrac{1}{\alpha_n}+\dfrac {1}{\beta_n})$

Hi, I know what roots are and how to find them but I don’t know why they are so important. What is that makes the points where a function become zero so important? I saw a similar post on this topic, but it talks about roots from an optimization point of view. However, finding roots...

Homework Statement Hello everyone. My task is to find the largest positive root in a specific interval of a function using the bisection method, Newton-Raphson method, and secant method. I've written code for all three of these methods, but the only way I can find all of the roots is to hard...

Homework Statement Hello everyone. My task is to find the largest positive root in a specific interval of a function using the bisection method, Newton-Raphson method, and secant method. I've written code for all three of these methods, but the only way I can find all of the roots is to hard...