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

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7
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0answers
60 views

Replacing numbers by roots of quadratic

We have $10$ numbers in the interval $(0,1)$, not necessarily distinct. At any moment, we can choose two of them, $a$ and $b$. If the quadratic $x^2-ax+b$ has two (possibly identical) real roots, we ...
4
votes
4answers
153 views

Solutions of $z^6 + 1 = 0$

Solve: $$z^6 + 1 = 0$$ That lie in the top region of the plane. We know that: $$(z^2 + 1)(z^4 - z^2 + 1) = 0$$ $$z = -i, i$$ We need to solve: $$((z^2)^2 - (z)^2 + 1) = 0$$ $$z = \frac{1 \pm ...
1
vote
3answers
59 views

A quadratic polynomial is nonnegative for all $x$ if and only if the discriminant is nonpositive

Show that if $a>0$ the inequality $ax^2+2bx+c\ge 0 $ for all values of $x$ if and only if $b^2-ac\le 0$. I tried to prove it by: $ax^2+2bx+c≥ b^2-ac$. Used partial derivatives with respect to ...
2
votes
1answer
43 views

Is the simplest form of a quadratic equation factored form or standard form?

I've done a bit of research about what defines simplest form, but I could not find a clear answer. Suppose we had to choose: $$(x - 4)(x + 2) \quad\text{or}\quad x^2 - 2x - 8$$ A question asked me, ...
1
vote
3answers
35 views

Quadratic formula in double inequalities

I have the double inequality: $-x^2 + x(2n+1) - 2n \leq u < -x^2 + x(2n-1)$ and I am trying to get it into the form $x \leq \text{ anything } < x+1$ Or at least solve for x as the ...
1
vote
2answers
46 views

Avoiding extraneous solutions

When solving quadratic equations like $\sqrt{x+1} + \sqrt{x-1} = \sqrt{2x + 1}$ we are told to solve naively, for example we would get $x \in \{\frac{-\sqrt{5}}{2},\frac{\sqrt{5}}{2}\}$, even though ...
1
vote
2answers
23 views

Factoring Quadratic

I have used the substitution P = dy/dx to solve a first-order D.E of degree 4, so I got this: I have to show that the above statement can be written as: I tried to factor out first by taking p a ...
1
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0answers
26 views

Show Equivalence of Binary Quadratic Forms

I've been stuck on these two problems from my problem set for quite a while. Any help would be appreciated! 2)Suppose that $ax^2 + bxy + xy^2$ is equivalent to $Ax^2 + Bxy + Cy^2$. Show that $gcd ...
3
votes
3answers
33 views

How to rearrange a quadratic into its factorized form?

Like the title says, I'm a bit confused about how the smart people of past centuries figured out that the quadratic: $$ ax^2+bx+c = a(x-x_1)(x-x_2). $$ The book I have at hand shows how to do it the ...
4
votes
2answers
444 views

Solving an equation with exponentials

$$2^x+4^x+12=0$$ How exactly am I supposed to solve this? Am I supposed to get $x$ alone or solve it another way?
0
votes
3answers
47 views

How do you reverse $\frac{100n(n+1)}{2}=c$ to find n given c?

I'm developing a game where the character experience needed by level is given by Gauss' formula multiplied by 100: $ \dfrac{100\mathrm{level}(\mathrm {level}+1)}{2}$. So the experience table is ...
2
votes
2answers
29 views

how to find A in quadratic projectile motion

what would the standard form be for this question? During a drumline performance, a drummer throws his drumstick with an upward velocity of 32 feet per second. if the drummer releases and catches the ...
1
vote
0answers
26 views

Real world application of slanted conics (parabolae especially)

I am writing a report on slanted conics of the form $$(x-h)^2+(y-k)^2= \dfrac{d}{\sqrt h}$$ Where $(h, k)$ is the focus, and $d$ is the directrix. Are there any real world applications for slanted ...
1
vote
1answer
15 views

If $a_0,a_1,a_2 \cdots a_{99} \in R$ and $f(x) =x^{100}+a_{99}x^{99}+a_{98}x^{98} +\cdots +a_0$ be such that $|f(0)|=f(1)$..

Problem : If $a_0,a_1,a_2 \cdots a_{99} \in R$ and $f(x) =x^{100}+a_{99}x^{99}+a_{98}x^{98} +\cdots +a_0$ be such that $|f(0)|=f(1)$ and each root of f(x) =0 is real and between 0 to 1. If product ...
0
votes
0answers
27 views

How to find the Coefficient of the Quadratic Term?

Given $4x^3 +bx^2+cx+d$ and two roots of this cubic function $(0,0)$ and $(2,0)$ Find the coefficient of the quadratic term? When I first read this I had no idea how to solve this and still ...
1
vote
0answers
24 views

All solutions to Quadratic matrix polynomials

I am after two things: 1- algorithms for finding all solutions of possibly large quadratic matrix equations of the form $AX^2+BX+C=0$ 2- (if possible) software implementing the algorithms ...
1
vote
1answer
36 views

Vertex Equation of an inverse quadratic function.

I'm working on a graphing web tool using JSXGraph, The user should be able to draw different functions. I was able to allow the user to draw quadratic functions by creating the vertex of the function ...
0
votes
2answers
16 views

What is the process of expanding quadratic equation

I am currently doing a math problem: $(a-b)(a^2+ab+b^2)$ However, I am not sure how I can actually expand this problem Do I multiply $(a-b)$ with each individual item within the other bracket?
2
votes
2answers
33 views

Quadratic equations: Why does factoring by grouping work?

We are learning factoring by grouping - The teacher explained the process but didn't explain the logic behind it. You need to multiply the coefficient on the x-squared term by the constant to get a ...
0
votes
1answer
28 views

Quadratic equation formula for a,b,c from 3 points

I can solve for a, b, c given three points for a parabola for example (1,1)(2,4)(3,9) but i need to create a program which returns a,b,c in the form: ...
2
votes
1answer
26 views

What is the solution for this quadratic program?

Given scalars $p_1\geq p_2\geq \cdots \geq p_r > 0$, can we find a solution for following problem? \begin{align} \text{minimize} & & & \sum_{j=1}^{r} p_j (1-t_j)^2 \\ \text{s.t.} \\ ...
0
votes
2answers
29 views

Quadratic Equations GRE Quants

It would be very useful if someone can give me an answer to this question with a proper explanation. One of the factors of the equation $x^2 +9x + c$ is $(x+11)$, where $c$ is a constant. Which of ...
0
votes
2answers
19 views

How to form a quadratic equation with real coefficients if $x_1=4-7i$?

Why is the quadratic equation $x^2-8x+65=0$? I tried to find $p$ and $q$ to form the equation but i need $x_2$ because: $$p=-(x1+x2)$$ $$q=x_1*x_2$$ so $x2=$?
2
votes
4answers
74 views

What does 'express in terms of $x$' mean?

For the following question : $f(x) = 2x^2 + 4x $ It asks me to express the following in terms of $x$: $f(-2x)$ What does the question mean by this? Does it mean make $x$ the subject?
9
votes
0answers
26 views

Intersection of two hyperplanes

$G$ and $H$ are hyperplanes in $\mathbb{P}_n$ with coordinates $g=(g_0, \ldots, g_n)$, $h=(h_0, \ldots, h_n)$. How can I find a symmetric matrix $A_Q$ of a quadric $Q$ with $ Q = G \cap H$, where ...
1
vote
1answer
20 views

max and minimum qudratic function problem

A piece of wire $20$ metres long is cut into $2$ pieces and each piece is bent to form a square. Determine the length of the two pieces so that the sum of the areas of the two squares is a minimum. ...
15
votes
1answer
264 views

An interesting table of Prime Generating polynomials similar to $n^2+n+41$?

Here is some data on quadratic prime generating polynomials of a particular form. Kindly look at the questions given below it. $$\begin{array}{cccc} \text{#} & P(n)=an^2+bn+c\,; & d = ...
0
votes
2answers
35 views

Factorize this polynomial $ax^2+bx+c$ into factors of the first exponent in the cases when D>0, D=0

The previous request was to prove the identity $ax^2+bx+c=a[(x+(b/2a)^2-(D/4a^2)]$, where $D=b^2-4ac$ And I proved it from the left to the right, which means I managed to express $ax^2+bx+c$ as ...
0
votes
1answer
15 views

Finding the y-vertex of a function and X2.

I am trying to solve the following exercise: The graph of the fuction $y=-2x^2+bx+c$ passes through the point (1,0) and has as its vertex the point (3,S). What is the value of s? Options: A -5_____ ...
0
votes
1answer
36 views

Discriminant of Quadratic with circle

The circle $x^2 + (y - c)^2 = r^2$, where $c > 0$ and $r > 0$, lies inside the parabola $y = x^2$. The circle touches the parabola at exactly two points located symmetrically on opposite sides ...
0
votes
0answers
15 views

Making equations for parabolas

When using the equation $y=a(x-s)(x-t)$ you are given the zeros of the quadratic are $0$ and $6$ and the minimum value is $-9$. What does the equation of the parabola look like?
0
votes
1answer
21 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 ...
0
votes
1answer
18 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 ...
0
votes
1answer
26 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 ...
1
vote
1answer
37 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 ...
1
vote
1answer
27 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.
0
votes
4answers
54 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 ...
0
votes
2answers
46 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 $$-0.0062\le\tau_1\le0.499$$ $$-0.479\le\tau_2\le0.0262$$ $$\tau_1+\tau_2\le0.02$$ ...
1
vote
5answers
151 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.
0
votes
2answers
30 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 ...
2
votes
4answers
37 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 ...
2
votes
1answer
21 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 ...
2
votes
3answers
21 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 ...
0
votes
1answer
75 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 ...
0
votes
1answer
36 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 ...
0
votes
1answer
26 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.
0
votes
0answers
25 views

Finding the domain and range of a function

$$F(x)=\frac {x^2+ax+1}{x^2+x+1}$$ Find the complete set of values of 'a' such that $F(x)$ is onto And f(x) maps from real numbers to real numbers.
1
vote
1answer
43 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 ...
1
vote
4answers
48 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?
0
votes
0answers
32 views

Trouble simplfying quadratic indentity

I'm struggling to follow the derivation below: $u_j=\frac{\sigma^4+ \theta^2\delta_j^2\alpha_j^2+2\theta\sigma^2\delta_j\alpha_j}{\sigma^2+\delta_j^2\alpha_j^2} \leq \frac{\sigma^4+ ...