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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3
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5answers
835 views

Where did $-4x$ come from?

I'm going over my quadratic equations for the ACT and I came across this quadratic: $$(x – 2)^2 – 12$$ My teacher said we could have factored it out into this: $$x^2 – 4x – 8$$ But I just don't ...
-1
votes
3answers
234 views

Finding parametric distance on quadratic curve from given $(x,y)$ point

I want to get the parametric distance (the "$t$" value) at a location on a quadratic Bezier curve, given the "$x$" and "$y$" coordinates of the point. I have start point, end point and control point ...
0
votes
3answers
42 views

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 ...
4
votes
2answers
210 views

Solving awkward quadratic equation to obtain “nice” solution.

I would like to solve the following quadratic equation to get a "nice" analytic solution for $\rho$. $\rho^2(r\sin\theta-2nr^2)+\rho(2nr^3-2r^2\sin\theta-2\sin\theta+2nr)-2nr^2+3r\sin\theta=0$ where ...
1
vote
4answers
137 views

If $ w_1 $and $ w_2 $ are the roots of $w^2-2sw+t=0$, then $|w_1|=|w_2|\iff 0<s^2/t\le1$

Let $s$ and $t$ be two complex numbers not equal $0$, $ w_1 $and $ w_2 $ are solutions to this equation $$w^2-2sw+t=0$$ How can prove this equivalence ? $$|w_1|=|w_2|\iff 0<\dfrac{s^2}{t}\le1$$. ...
4
votes
2answers
96 views

Solution for this Logarithmic Equation

Recently I was going through a problem from the book Problems in Mathematics - *V Govorov & P Dybov* . $$(x-2)^{\log^2(x-2)+\log(x-2)^5-12}=10^2\log(x-2)$$ I tried solving by first considering ...
0
votes
3answers
34 views

finding an quadratic equation by the roots & another equation?

I am new to this site & doesn't know any rules & regulations. So sorry if I am doing any mistake. the question is stated as follows. I. $\alpha$ and $\beta$ are the roots of the equation ...
1
vote
2answers
46 views

Setup Quadratic Word Problem

I need help setting up this quadratic word problem, I have no idea where to start. Among all pairs of (real) numbers whose sum is 17, find a pair whose product is as large as possible. What is the ...
4
votes
0answers
55 views

Computing question: A quadratic which gives primes [closed]

This is about Project Euler Problem 27. The question is: Considering quadratics of the form $n^2 + an + b$, where $\lvert a \rvert < 1000$ and $\lvert b \rvert < 1000$ Find the product ...
0
votes
3answers
48 views

how to solve this quadratic equation

$n^2-4n+2=0$ I have tried many things for this but I cant resolve the roots here $n$ should be a positive whole number as it stands for time.
0
votes
1answer
34 views

Quadrature of $\int_{-2}^2 e^{-x} f(x) dx$ by $\alpha_0f(-1) + \alpha_1f(0) + \alpha_2f(1)$

I am looking for an approximation $$\alpha_0f(-1) + \alpha_1f(0) + \alpha_2f(1)$$ of $$\int_{-2}^2 e^{-x} f(x) dx $$ that is exact for polynomials $f$ of degree 2. My first idea is to solve these ...
1
vote
1answer
122 views

Quadratics Word Problem

The path of a football flying through the air can be modelled by a quadratic equation. The football reaches the ground after 12 seconds in flight and is kicked from a height of 1 meter. The parabola ...
1
vote
3answers
40 views

Given the equation of the parabola, for what values of $t$ will the quadratics have $0$, $1$, or $2$ solutions?

Given the equation of the parabola, for what values of $t$ will the quadratics have $0$, $1$, or $2$ solutions? Equation: $$0 = 3x^2 + tx + 10$$ Can you please explain the answer in simple terms, ...
0
votes
5answers
42 views

Smallest value of function on a line

Problem : If the point $(\alpha, \beta)$ lies on the line $2x+3y=6$, the smallest value of $\alpha^2+\beta^2$ is (a) $36/13$ (b) $6\sqrt{13}/13$ (c) $6$ (d) $13$ Solution : Since ...
0
votes
1answer
63 views

Modulo Quadratic Polynomials

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.
2
votes
5answers
209 views

Show that this expression is a perfect square?

Show that this expression is a perfect square? $(b^2 + 3a^2 )^2 - 4 ab*(2b^2 - ab - 6a^2)$
2
votes
4answers
344 views

What is a complex constant and how do I use it?

I have a question I am trying to understand: "Let $b$ and $c$ be complex constants such that $z^2+bz+c=0$ has two different real roots. Show that $b$ and $c$ are real." My biggest problem here is ...
1
vote
0answers
48 views

Quadratic Congruence in $\mathbb Z/2^n \mathbb Z$

Given the congruence $ax^2+bx+c \equiv 0 \pmod {2^n}$, how precisely does one go about finding its roots? I'm comfortable with quadratic congruence mod n with n odd, but 2's lack of a multiplicative ...
1
vote
3answers
42 views

How do I solve: $6(x^2+2)<17x$

How do I solve this kinds of inequality. I can do it if all the 'x' is in one side. However, this one have x at both sides of the equation. And we don't know whether it's a positive or negative value. ...
3
votes
1answer
107 views

Solution to a System of Quadratic Equations

Problem: Solve for the values of a, b Equation 1: $$(x_1-a)^2+(y_1-b)=r^2$$ Equation 2: $$(x_2-a)^2+(y_2-b)^2=r^2$$ Where, $x_1, x_2, y_1, y_2$ and $r$ are all constant values For the ...
2
votes
4answers
61 views

How to check, when/whether a quadratic equation has iteger roots

I am familliar with the Jacobi symbol and thus can check for an equation, whether it has solutions in a given $\mathbb{Z}_n$. But how do I check for integer solutions in general? In perticular, I ...
2
votes
2answers
174 views

How to implement a numerically stable solution of a quadratic equation?

Solving $a x^2 + bx +c=0$ for $x$ gives $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ for $a \ne 0$. But for $a = 0$ we get $x=-\frac{c}{b}$. How to implement a numerically stable algorithm computing ...
3
votes
3answers
179 views

Can this result be simplified?

I was trying to solve the value of $x$ in $x^4-4x^2-2 = 0$ in terms of radical. The answer I got is $x=\sqrt{2+\sqrt{6}}$. How can this value be simplified even more, while still expressing it in ...
1
vote
1answer
74 views

How to solve $\frac{2x+1}{2x-3}+\frac{7x\:}{9-4x^2}=1+\frac{x-4}{2x+3}$ for $x$?

Can somebody explain me this one! $\frac{2x+1}{2x-3}+\frac{7x\:}{9-4x^2}=1+\frac{x-4}{2x+3}$ My book says the answer is $x_1 = 0$; $x_2 = 6$. I tried to solve it and got stuck somewhere in: ...
0
votes
1answer
36 views

How to obtain root of this quadratic equation

I got this quadratic equation in a problem $T^2 - 2T - 40 = 0$, but i am unable to find the roots of this equation. Is any other concept hidden in this equation? Please help. I am basically a Bipc ...
2
votes
3answers
48 views

can i solve this quadratic equation this way

I was basically doing a physics problem and came across this equation in midway $\dfrac{2n-1}{n^2} = \dfrac{11}{36}$ then I equated $2n-1 = 11$ and $n^2 = 36$ and the value of $n$ which I got is ...
2
votes
2answers
64 views

Solving $x^2+y^2=a$

I'm looking for a lesson or a link to a well explained proof on how to solve equations of the form $$x^2+y^2=a$$ where $x,y,a\in\Bbb N$ are all natural numbers. Regards
1
vote
1answer
700 views

How to find the solution of a quadratic equation with complex coefficients?

I know how to find the solution for a quadratic equation with real coefficients. But if the coefficient changes to complex numbers then what is the change in the solution? Want an example of such ...
0
votes
5answers
63 views

Solving a quadratic trigonometric equation?

The equation is $6 \cos^2x+\cos x=1$, My work: $6x^2+x-1=0$ $(3x-1)(2x+1)$ $3x-1=0 ∨ 2x+1=0$ $x=\frac{1}{3} ∨ x= \frac{-1}{2}$ But I do not know how to progress further.
0
votes
4answers
493 views

Using sum/product of quadratic roots to solve cubic equation

Given $\alpha$ and $\beta$ are the roots of the quadratic equation $6x^2 + 2x - 3 = 0$, how do I find the value of: $$ \alpha^3 + \beta^3 $$ and: $$ \frac{1}{\alpha^3} + \frac{1}{\beta^3} $$ ...
1
vote
5answers
99 views

Distribution of integer solution pairs (x,y) for $2x^2 = y^2 + y$

I am looking for integer pairs $(x,y)$ that respect $$2x^2 = y^2 + y$$ For example $(6,8)$ is a solution for that. Simple solution is to enumerate on $x$ or $y$ and test if the corresponding ...
0
votes
1answer
46 views

Find domain of function with quadratic numerator algebraically

I'm stuck on this problem: $$f(x) = \frac{x^2 -4}{x}$$ I need to determine why this function's domain is not: $$\{x|x \neq \pm 2\}$$ All of the examples that I've seen have the quadratic in the ...
0
votes
2answers
41 views

How to prove one of these equations has real roots?

Question: If $\;a,\, b,\, c,\, d \;$ are real and $\;ac = 2( b + d) $ ,then show that at least one of the equations: $\; x^2 + ax + b = 0 \;$ and $\; x^2 + cx + d = 0 \;$ has real roots. I've ...
0
votes
3answers
42 views

Quadratics with unknowns

If $5x^2 – t = 4x$, and $x$ and $t$ are both positive real numbers. What is $x$ equal to? How do you find $x$? Is there a specific formula?
1
vote
2answers
256 views

Finding the probability an equation has real roots.

If $Q$~UNIF$(0,3)$, find the probability that the roots of the equation $g(t)=0$ are real, where $g(t)=4t^2+4Qt+Q+2$. There was a similar question asked that I looked at, but I am still a little ...
0
votes
1answer
27 views

Quadratic factor to complex numbers

How to convert this quadratic factor to complex number form? (With steps please) Reference: $Z = a + bi$, $i = \sqrt{-1}$ $$-3 + \frac{\sqrt{-12}}{2}$$ Thanks!
0
votes
1answer
34 views

Common root of quadratic.

If the quadratic equations, $x^2+bx+c=0$ & $bx^2+cx+1=0$ have a common root. Prove that either $b+c+1=0$ or $b^2+c^2+1=bc+b+c$. Please also explain What should be the logic / approach we should ...
1
vote
1answer
43 views

system of two quadratic equations with two variables

Is there a general way to solve exactly a system of this shape (the $a_i$ are constants): $$\begin{array}{cc}a_1x^2+a_2x+a_3y^2+a_4y+a_5=0\\ a_6xy+a_7x+a_8y+a_9=0 \end{array} $$ It comes from a ...
0
votes
2answers
81 views

Is this a quadratic equation?

I have this equation: $-19y^2 + 5y\sqrt{y} + 10y + 12 = 0$ I'm stuck with finding $y$. I tried numerous adjustments like this for example: $y*(-19y+5\sqrt{y}+10)+12 =0$ but it didn't get me any ...
1
vote
1answer
108 views

When is $(x+3)^2$ equal to $x^2 +9$?

http://matheducators.stackexchange.com/a/1400/775 Someone commented that the equation in the above answer might sometimes be correct after I commented a correction (feel free to rewrite it ...
2
votes
4answers
376 views

Finding the two digit number.

A two digit number is such that the product of its digit is 18.When 63 is subtracted from the number , the digits interchange their places.Find the number.
2
votes
1answer
95 views

Nonlinear first order ODE with quadratic in the derivative

This equation shouldn't be so hard, and yet I'm stymied. $$ \left( \frac{dw}{dz} \right )^2 + \alpha \frac{dw}{dz} + w \beta = 0 $$ with $w(0) = w_0>0$ $w(L) = 0$ for some known L and ...
1
vote
3answers
144 views

Determine the value(s) of the ratio $x:y$ if $2x^2-xy-3y^2 = 0$

I was asked to determine the value(s) of the ratio $x:y$ if $2x^2-xy-3y^2 = 0$. I didn't know what this meant, so I just solved the equation to get $x = -y$ or $x = \frac{3}{2}y$. What does the ...
0
votes
1answer
67 views

Type of this Conic section

I want to determine, to which type the following Conic sections belong to: $$ \begin{align} \textrm{(i)}&\quad-8x^2+12xy-6x+8y^2-18y+8=0\\ \textrm{(ii)}&\quad5x^2-8xy+2x+5y^2+2y+1=0 ...
0
votes
1answer
32 views

Find roots of binomial expression by replacing some variables?

So we have the binomial expression * I might be not using the correct term,english isnt my first language* $$ \left[1- \frac 34e^{-j2\pi\cdot f} + \frac 18e^{-j4\pi \cdot f} \right]$$ How do I find ...
0
votes
4answers
41 views

Finding the three unknowns

Can someone show me the steps to finding the three unknowns of these two equations. $$-a-bx+cx^2 = x^2+2x+1$$ The answers are $a=\ ...\ $, $b=\ ...\ $, and $c=\ ...$ , but I can't see how they ...
0
votes
1answer
67 views

Homework: canonical form of quadratic form

X=(x,y,z) Q(X) = $x^2 + 4xy + 6xz + 3y^2 +8yz +5z^2 $ I got by using completing the square method: Q(X) = $(x+2y+3z)^2 - (y+2z)^2$ so as I learned now I do: $u = x+2y+3z$ $v = y+2z$ $w = 0 $ ...
0
votes
1answer
19 views

system of equations solving with only that information

Hi would would I go around to solve the following, there is no other information stat is given other than the fact that i have already expanded this from this $(25-y)(x+8)=523$ $25x-8y=323$
5
votes
2answers
689 views

Find the value of a + b + c + d

Let $a$ and $b$ be the roots of the equation: $x^2 - 10cx - 11d = 0$ where $c$ and $d$ be the roots of $x^2 - 10ax - 11b = 0$. Find the value of $a+b+c+d$, assuming that they all are distinct. I ...
-1
votes
1answer
49 views

Expressing quadratic equation in terms of its roots

For a quadratic equation, $ax^2 + bx + c = 0$, why is $ax^2 + bx + c = a(x-\alpha)(x-\beta)$ where alpha, beta are the roots of the equation? Why not just $(x-\alpha)(x-\beta)$?