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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5answers
63 views

For what real number $c$, this equation has exactly three solutions?

For what real number c does the equation $|x^2 + 12x + 34| = c$ has exactly three solutions?
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3answers
42 views

Finding dimensions using quadratic formula

A 52 m long fence is constructed on three sides of a housing block with area 240 m^2. The fourth side facing the road is left open. Find the dimensions of the block. Also here's another question I ...
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1answer
30 views

Quadratics: Solve for B and C.

If I know that the points (-1,6) and (2,3) are on the graph of the quadratic function f(x) = 2x^2 + bx + c, how do I determine b and c? Thanks to anyone who helps.
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1answer
22 views

Quadratic Functions: Determine the value of b

I'm having trouble with this question and I'm not sure what to do. Would appreciate any one who helps me out. Question: The point $(-2,1)$ is on the graph of the quadratic function: $f(x) = -x^2 + bx ...
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2answers
58 views

Quadratic Formula for complex variable with real coefficients

I have been upto proving the following $$(\forall x\in \mathbb C, ax^2 + b x + c = 0) \land(a\neq0)\Leftrightarrow (x = {\frac {-b \pm \sqrt{b^{2}-4ac}} {2a}})$$ Due to equality we need to proof bi ...
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2answers
44 views

Consider a quadratic equation

$x^2-2(a+1)x+3a+2=0$ Where a is any real number. Find all values of a so that the equation has two distinct real roots. I tried solving for a by using the inequality $b^2-4ac\gt0$... but when i ...
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6answers
1k views

what would be the way to solve a system of equations like this one?

Solve: $xy=-30$ $x+y=13$ {15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?
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1answer
42 views

Quadratic problem

NASA launches a rocket at $t = 0$ seconds. Its height, in meters above sea-level, as a function of time is given by $h(t) = -4.9t^2 + 37 t + 101$. Assuming that the rocket will splash down into the ...
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2answers
48 views

Equation with the discriminant

I'm stuck on my maths homework, and would appreciate help. The question is: Show that if the equation $(m + n)x^2 - 2mnx - (m-n) = 0$ has equal roots, then $$m^2 = \frac{n^2}{1-n^2}$$ I've worked ...
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1answer
44 views

What is the relationship between three points on a quadratic curve and the curves coefficients?

In other words, is there a formula to get the coefficients a,b and c in terms of three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3, y_3)$? I am asking this because I have a linear algebra problem that ...
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0answers
65 views

Question on simple quadratic word problem regarding weekly revenue when price of merchandise is lowered

A store owner sells headphones at 24 dollars a piece with roughly 1000 sold per week. The store owner finds that for every 1 dollar decrease on the price per headphone he sells 100 more headphones per ...
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2answers
43 views

Can one solve this system of qudratic equations unambigiously?

Given the parameters $p_1, p_2, p_3$ I want to know if the following system can be solved: $p_1 a + p_2 c + ef = 0\\ p_1 b + cd + p_3 f = 0\\ ab + p_2 d + p_3 f = 0\\ p_1^2 +a^2+b^2=1\\ ...
2
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2answers
77 views

If the equation $|x^2+bx+c|=k$ has four real roots …

If the equation $|x^2+bx+c|=k$ has four real roots then which of the following option is correct : (a) $b^2-4c >0$ and $0<k<\frac{4c-b^2}{4}$ (b) $b^2-4c <0$ and ...
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3answers
35 views

Tangent of a rational function

I'm wondering if someone could help me solve a little conundrum I've been having. I'm trying to find the equation of a line with slope $-1$ tangent to the curve of a rational function. The function is ...
13
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10answers
4k views

Taking Calculus in a few days and I still don't know how to factorize quadratics

Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ...
2
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2answers
68 views

Quadratic Prime

We had received some questions on Quadratic equations, But I wasnt able to do one. Here it goes: Let $a,b$ be natural numbers $a>1$. Also, $p$ is a prime number. If $ax^2+bx+c=p$ for 2 distinct ...
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2answers
27 views

How to Find The Roots of this Quadratic Given Sum & Product

My question is: The sum of the roots of a quadratic is $55/72$, and the products of the roots is $-25/12$. Find the roots. How I'm trying to do it so far: (Also, please correct my thought process if ...
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1answer
45 views

Solve the equation $x^2+bx+c+t\log(x)=0$

There is a explicit formula to solve the equation $x^2+bx+c+t\log(x)=0$ with the constraint $x>0$?
2
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5answers
72 views

Algebraic process to find numbers so that $xy=45$ and $x+y=18$

Can someone help me with the following question? The sum of two numbers is $18$ and their product is $45$. Find the numbers. I know that the answer is $15$ and $3$. But how do I find that answer ...
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1answer
41 views

Quadratic equations - finding value of x

Question: Solve the equation $$\frac{2x}{3x^2 - x+2} - \frac{7x}{3x^2 + 5x+2} = 1$$ I attempted to split the two quadratic equations into their roots, however, was unable to do so. Then I tried to ...
2
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4answers
56 views

Quadratic equation - solving for x

Question: Solve the equation $$(x+2)(x+3)(x+8)(x+12) = 4x^2$$ I tried to solve the equation by expanding the LHS and then equating it to the RHS, but that just doesn't seem to be feasible. I am ...
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7answers
316 views

How to solve the quadratic equation $x^2-1=2$?

Solve $x^2-1=2$ I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.
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3answers
75 views

How do I factor this quadratic?

I'm going through the AoPS Algebra book, and I'm on the quadratics section. I'm given this challenge question: $ \displaystyle 2x^2 + 7x(\sqrt{3}) + 9 = 0$ And I have to solve for ...
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2answers
63 views

Using factoring to solve the equation $(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$

Solve for all values of $r$: $$(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$$ I'm not sure how my thinking isn't really correct here. I know this all seems very elementary and such, ...
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1answer
43 views

Quadratic equations ..

The set of non-zero values of k such that the equation $|x^2-10x+9| =kx$ is satisfied by atleast one and atmost three values of x, lies in _. The answer is $(-\infty, -16] \cup [4 , \infty) $. How ...
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4answers
70 views

Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
2
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2answers
34 views

Quadratic equation not equal to zero (solving a matrix with a parameter)

I came across this in my matrix module, learning about number of solutions when the matrix has parameters. $$ \left[ \begin{array}{ccc|c} 1 & ...
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3answers
66 views

How to solve this word problem on the topic of quadratic equations?

The maths teacher of Mumbai is transferred to another school. The students of Class 10 decided to buy a book for 360 rupees(currency) as a gift for her. On the farewell day, 4 students did not turn up ...
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1answer
41 views

Difference between Half Quadratic vs Quadratic

Half quadratic minimization/penalty/optimization, I am unable to find any related material/resources. If anyone can point to some useful resources, it will be great
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1answer
24 views

Check if a point is within a quadratic surface (with arbitrary rotation)

Is there a general way to check whether a point is on a quadratic surface given that the principal axes do not need to coincide with the coordinate axes and that the quadric's centroid does not need ...
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2answers
42 views

Can I perform the quadratic formula on polynomial with complex coefficient?

2 weeks ago, we had a Math test on complex number. One of the question was: Let $z=x+iy$ be a non-zero complex number, where $x,y \in \mathbb{R}$. Given that $z+\frac{1}{z} = k$, where $k$ ...
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1answer
61 views

Cross section of parabolic satellite in Quadratic Functions

A parabolic satellite dish has a cross section that can be modelled by the equation $$y = 0.05\,x^2.$$ While still in the shipping yard, the dish fills with rain. The rain forms a circular puddle with ...
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3answers
83 views

Absolute value quadratic inequalities not the usual?

$ | -x^2 + 6x | \gt 13 $,for example. I would start off solving $ -x^2 + 6x = \pm 13 $ and either get 4 solutions, 3 solutions or two simply do the the nature of the graph. Without knowing if the two ...
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5answers
802 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 ...
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3answers
146 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 ...
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3answers
40 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 ...
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2answers
202 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 ...
3
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2answers
79 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 ...
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3answers
30 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
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2answers
34 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 ...
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0answers
51 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 ...
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3answers
45 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.
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1answer
83 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
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3answers
38 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, ...
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5answers
40 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 ...
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1answer
53 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.
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5answers
205 views

Show that this expression is a perfect square?

Show that this expression is a perfect square? $(b^2 + 3a^2 )^2 - 4 ab*(2b^2 - ab - 6a^2)$
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4answers
150 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 ...
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0answers
46 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
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3answers
41 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. ...