Questions on quadratic equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-h)^2+k$ or $y=a(bx+c)(dx+e)$.

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Factorizing Given Problem

I have searched through various site's and forums but couldn't find the answer to my problem, $$z^2-\frac{1}{2}z-\frac{1}{4}=0$$ How will you factorize this As I can't find $2$ numbers that give me ...
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2answers
40 views

Finding out the quadratic equation using Vieta? [closed]

So I have the solutions to a quadratic equation: $x_1=\frac{-3}{2}$ $x_2=\frac{1}{4}$ $x^2+px+q=0$ (Just telling you as I've seen many people using other letters for the variables) I tried ...
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1answer
23 views

What can the “Product of Roots” be used for in quadratic form?

If I have a linear function and some kind of quadratic in x and y ie: $x^2+xy+y^2=1$ that share two roots, then I can substitute that linear function into the quadratic expression and use the Sum of ...
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0answers
24 views

Linear constraints in Quadratic equation

I have been going through this paper, and wish to implement the same algorithm in java. I have also managed to write equivalent code for the same, but I have not completely understood the mathematics ...
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1answer
27 views

Grade 10 Quadratic equation

This was on my year 10 maths test and I gave up with 40 mins to complete: Basically you were given the coordinates: y intercept : (0,10) 1 x intercept: (10,0) and y value of the vertex: +15 Can ...
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1answer
57 views

Writing a equation in vertex form with an axis of symmetry, maximum height, and a point that it crosses

Suppose a parabola has an axis of symmetry of $x = -7$, a maximum height of $4$, and passes through point $(-6, 0)$. Write the equation in vertex form. Here's what I got: $y = -(x + 7)^2 + 4$ The ...
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4answers
2k views

Why can we prove mathematically that a formula to solve an (n+5) order polynomial does not exist?

I understand that the quadratic equation can solve any second order polynomial. Furthermore, equations exist for polynomials up to fourth order. However, without a graduate level degree and a deep ...
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1answer
61 views

Solve quadratic equations modulo prime powers

To find if $x^2 = a \mod p$, I use the Tonelli-Shanks algorithm. However, how do I find the roots for $x^2 = a \mod p^t$, if I have solved the previous equation? Thanks
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1answer
43 views

Polynomial function question

If $f(x)$ is equal to $\frac{1}{x^3 + 3x^2 + x}$, find the smallest value of $n$ for which $f(1) + f(2) + ... F(n) = \frac{503}{2014}$. I tried noting that first initial values of f sum to ...
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1answer
30 views

Determine all values of n such that this quadratic

Determine all values of $n^2 + 19n + 99$ is a perfect square. I tried setting some square $b^2$ equal to the following, and then factoring as a Diophantine equation with $2$ variables... Didn't work.
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4answers
59 views

Quadratic equation $3x^2 + x - 2 = 0$

I have $3x^2 + x - 2 = 0$ and the answers are supposed to be $-1$ and $2/3$. It was in the quadratic formula chapter so I tried to use that but since the middle x is only 1 for a coefficient, it ends ...
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2answers
56 views

Quadratic programming for special equation issues

My problem is how to find $\tau_1$ and $\tau_2$ s.t maximize the objective function is $$E=M-\alpha V$$ subject to $$-0.0062\le\tau_1\le0.499$$ $$-0.479\le\tau_2\le0.0262$$ $$\tau_1+\tau_2\le0.02$$ ...
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2answers
45 views

Solving Complex Quadratic equations

After working a few exercises on the topic, the questions become progressively harder. In this particular exercise I was asked to solve the equations. However I can't quite seem to break this problem ...
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5answers
170 views

completing the square to solve equation

Is it possible to use the method of completing the square to solve the equation $2x^2+18x+21=0$ ? I have problem with how to remove the negative sign on the right side.
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3answers
49 views

Where am I going wrong with this completing the square exercise?

I have been trying to learn some pre-calculus stuff in advance for next year (attempting a university paper) I am trying to solve a completing the square equation but can't see where I am going wrong ...
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2answers
48 views

Quadratic expression into postfix notation

I know generally how to convert an infix expression into a postfix expression; but I came lately across this quadratic expression: $\left(4y^2 + 2x - 1\right)$ that I had to convert into postfix and ...
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4answers
38 views

Solve $\frac{1}{2}kx^{2}-cx=\frac{1}{2}ky^{2}+cy$ for $y$

I have the equation: $\frac{1}{2}kx^{2}-cx=\frac{1}{2}ky^{2}+cy$, where $k$ and $c$ are arbitrary constants. How do I go about simplifying this and solving for $y$ in terms of $x$, excluding the ...
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3answers
40 views

solve quadratic equation

I'm trying to solve the following equation $2t^2 + t - 3 = 0$ I start by dividing by 2, $t^2 + \frac {t}{2} - \frac {3}{2} = 0$ Then I solve for t $t = - \frac{ \frac {1}{2} }{2} \binom{+}{-} ...
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2answers
21 views

Finding the other $x$-intecept of a quadratic equation $ax^2+bx+c=0$ when $a$ is unknown and one $x$-intercept is known

One of the $x$-intercepts of the function $f(x)=ax^2-3x+1$ is at $x=-1$. Determine $a$ and the other $x$-intercept. I happen to know that $a=-4$ and the other $x$-intercept is at $x=\frac{1}{4}$ but ...
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1answer
22 views

For $f(x) = ax^2 + bx +c$, why is it written $a(x-h)^2 + k$

I'm going to have to teach how to graph quadratic equations. Since we've already done a lot of work with the Quadratic Formula, the students are more or less familiar with the standard notation of a ...
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1answer
146 views

Quadratic equation, math olympiad question

So this is a 9-10th grade, math olympiad problem I found. Define the parabola $y=ax^2+bx+c$ such that $a,b,c$ are positive integers. Suppose that the roots of the quadratic equation $ax^2+bx+c=0$ are ...
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3answers
46 views

Conceptual problem in solving quadratic equation

The sum of all real roots of the equation $$|x-2|^2 + |x-2| - 2 = 0$$ is? I tried this problem by taking two cases $x<2$ and $x>2$ and solving the corresponding equations and I got $8$ as the ...
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3answers
26 views

Conditions on polynomials with common roots.

If one root of the equation $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ is common and $a \ne b$ then: The options are as follows: $$\begin{array}{ll} (A)\quad& a + b = 0\\ (B)& a + b = -1\\ ...
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1answer
81 views

Can a quadratic be solved with matrices?

The question, pure curiosity, is whether you can solve a quadratic with the use of matrices? And if yes, does that method also work for higher polynomials? Say for example I have a quadratic such as ...
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1answer
42 views

If $\alpha,\beta$ be the roots of $ax^2+bx+c=0 (a,b,c \in R)$

If $\alpha,\beta$ be the roots of $ax^2+bx+c=0 (a,b,c \in R) , \frac{c}{a}<1$ and $b^2-4ac <0$, $$f(n) \sum^n_{r=1} (|\alpha|^r +|\beta|^r)$$ then $$\lim_{n\to \infty} f(n) $$ is equal to ? Sum ...
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4answers
41 views

How do you find out the range of values when dealing with simultaneous equations?

Find the range of value for $k$ for which $kx + y = 3$ meets $x^2 + y^2 = 5$ in two distinct points. im so stuck can someone give me a clear guide to the correct method and answer, thank you
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1answer
31 views

Find the domain and range of a quadratic [duplicate]

$$F(x)=\frac {x^2+ax+1}{x^2+x+1}$$ Find the complete set of values of 'a' such that $F(x)$ is onto And f(x) maps from real numbers to real numbers.
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1answer
48 views

quadratic equation

A garden is in the shape of a rectangle, $20$m by $8$m. Around the outside is a border of uniform width and in the middle is a square pond. The area which is not occupied by either border or pond is ...
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1answer
43 views

Parametrization of $ax^2+bxy+c=0$

Can I just fix $y=t$ and use quadratic formula to get the rational points of the diophantine $$ax^2+bxy+c=0?$$ or is there another method? I feel like I am turning in circles with the quadratic ...
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3answers
57 views

quadratics equation tricky problem

I am confused with this question- if $ax^2+bx+c$ have no real roots then- $1+c/a+b/a$ is-- a. Positive b. Negative c. Zero d. Can.t say I tried attempting it as follows $b^2-4ac<0$ so ...
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4answers
51 views

quadratic equation: $5x^2 + 9x - 170 = 0$

I have a problem, my textbook says the solution of $5x^2 + 9x - 170 = 0$ is $5$ but the book didn't describe how it solved the equation. How can I solve this?
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0answers
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Solving matrix equation of the form $(AX)^2+(BY)^2=D$

Is there any method that can solve the matrix equation of the form $(AX)^2+(BY)^2=D$? $A$ and $B$ are matrices, $X$, $Y$ and $D$ are column vectors. (Solve for $X$ and $Y$) I originally have two ...
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1answer
66 views

Question about quadratic equation of complex coefficients.

Let $az^2+bz+c=0$ be a quadratic equation with complex coefficients $a,b,c$ and roots $z_1, z_2.$ If it is given that $|z_1|\not=|z_2|,$ how can I obtain the condition for this containing $a,b,c?$ ...
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2answers
471 views

How to solve this equation? Can I treat as a quadratic equation?

$$\ln(x+3)+\ln(x-4)=0$$ How to solve this equation? First removing the 'ln' from the equation and after making a quadratic equation and then solve the quadratic equation?
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1answer
43 views

If the roots of $x^2+x-1$ are $\alpha$ and $\beta$, find an $eq^{n}$ whose roots are $\alpha^{19}$ and $\beta^{7}$

If the roots of $x^2+x-1$ are $\alpha$ and $\beta$, find an $eq^{n}$ whose roots are $\alpha^{19}$ and $\beta^{7}$ My Procedure The roots are $$\frac{-b+\sqrt{b^{2}-4ac}}{2a}$$ and ...
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1answer
59 views

Non-standard quadratic matrix equation

I have an equation that looks like the following: $$ A\cdot\mathrm{diag}(x)\cdot x + B\cdot x + c = 0 $$ where $A, B, C \in \mathbb{R}^{n \times n}$ and $x, c \in \mathbb{R}^n$. $ x $ is unknown. ...
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1answer
43 views

List the elements of the set $\{X \in \mathbb Z \mid 4X^2 +11X = 0\}$ [closed]

I don't get this maths equation Can anybody explain it ? Thanks List the elements of the following set: $A=\{X \in \mathbb Z \mid 4X^2 +11X = 0\}$.
2
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1answer
26 views

Evaluating cubic roots of a quadratic

If $\alpha$ and $\beta$ are the roots of the quadratic equation $2x^2 + 4x -5 = 0$, evaluate $\alpha^3 + \beta^3$.. I know that $$\alpha + \beta = \frac{-b}{a}$$ and $$\alpha \beta = ...
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2answers
30 views

Factor the Quadratic

-16t^2+32t+20=0. How are you supposed to find -5 and positive 1 to put in the parenthesis? -4(2t-5)(2t+1)?
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0answers
23 views

About diagonalizing a matrix for a quadratic expression (with the goal of uncoupling mixed terms)

my question is originated from a physical problem. I will try to present the problem as simple as possible, but I fear it will still be long since I'm bad at expressing myself briefly. It starts with ...
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1answer
51 views

Solve system with different variables

I need to solve the system: $$x^2+2xy+y^2-1 = 0$$ where variable is $x$ AND $$x^2 + 2xy = 0$$ where variable is $y$. From the first Ι take discriminant, and end in one solution $x_1 = 1-y$ and ...
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3answers
60 views

Inequalities $- x^2 - (1/2) x - 5 < 0$ ; why is $x > 5/2$?

Question : $$\text{ find the set of values of }x \text{ for which } $$ $$10 + x - 2x^2 < 0$$ Answer : $$x < -2$$ $$x > 2\frac{1}{2}$$ EDIT - thanks for the responses. To try and ...
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1answer
99 views

Quadratic Irrationality of the Periodic points of the Gauss map

If $G:[0,1] \rightarrow [0,1]$ is the Gauss map which is defined as $$G(x) = \left\{\frac{1}{x}\right\} = \frac{1}{x} - \left\lfloor\frac{1}{x}\right\rfloor,$$ show that if $x$ is periodic of order ...
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1answer
50 views

Issue on proving quadratic formula

I have come across a stage of the proof: $$ \left(x+\frac b{2a}\right)^2=\frac{b^2-4ac}{4a^2}$$ How does $\left(x+\frac b{2a}\right)^2$ not equal $\pm x\pm \frac b{2a}$ when taking the square root?
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1answer
64 views

In what base does the equation $x^2 - 11x + 22 = 0$ have solutions $6$ and $3$?

If we have below equation and know that $6$ and $3$ are answers of this equation, how to obtain the base used in the equation? $$x^2 - 11x + 22 = 0$$ Partial result The base is not $10$. (Because ...
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3answers
40 views

Manipulate the Physics Equation $P = I^2R$ to get R by itself

Given that $P = (V^2 R_1)/(R_1 + R_2)^2$, manipulate the equation so that we get $R_1$ by itself and that we have a quadratic equation. Where $V, P, R_1$, and $R_2$, are variables. I'm stuck when I ...
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1answer
22 views

Inequalities and equations - creating sets from quadratic equations.

My question is just making sure that my working is correct and that I understand properly (self teaching, can get confused...) So question : Find the set of values for which $$x^2 -4x-12 < 0$$ ...
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2answers
28 views

What is the correct answer to this diffferential equation?

[Question] When solving the differential equation: $$\frac{\mathrm dy}{\mathrm dx} = \sqrt{(y+1)}$$ I've found two ways to express $y(x)$: implicitly: $2\sqrt{(y + 1)} = x + C$ or directly: $y = ...
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1answer
34 views

Quadratic equation form

I have the relation $u=\sqrt{(a_1+b_1t)^2+(a_2+b_2t)^2+(a_3+b_3t)^2} \tag 1$ I need to write $t$ as a function of $u$ ($t=f(u)$). How will I get that ? NB: $a_1,a_2,a_3,b_1,b_2,b_3$ are ...
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3answers
66 views

Quadratic equation $9x^2-37=6x$ using the quadratic formula

Quadratic equation using the quadratic formula $9x^2-37=6x$ So $9x^2-6x-37=0$ $A= 9$ $b=-6$ $c=37$ $\dfrac{-(-6) \pm \sqrt{ (-6)^2- 4(9)(37)}}{2(9)}$, $\dfrac{6 \pm \sqrt{36-1332}}{18}$, $\dfrac{6 ...