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

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691 views

EdExcel GCSE question about Hannah and the sweets: show that $n^2-n=90$

This is my reconstruction of the EdExcel GCSE question that has caused such a Twitter storm in the UK in the last 24 hours, along with its solution. Hannah has a bag containing $n$ sweets, 6 of ...
2
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2answers
42 views

Determinate set $A\subseteq\mathbb{R}$ so that for every $a\in A$ and every $x\in\mathbb{R}$ the condition $ax^2+x+3\ge0$ is valid

Quadratic function is always greater than $0$ if $$a>0$$ and $$D=0$$ Solving for $D$ we have $$1-12a=0\Rightarrow a=\frac{1}{12}$$ So, $$a\in[\frac{1}{12},+\infty)$$ Is this the only condition ...
2
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4answers
155 views

Show that $3$ is not a prime in $\mathbb Q[\sqrt{7}]$

Question: Show that $3$ is not a prime in $\mathbb Q [\sqrt 7] $. To show this, should I start by assuming that $3 = ab$ where $a$ and $b$ are integers in $\mathbb Q[\sqrt{7}]$ and then try to ...
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0answers
24 views

Convex quadratic problem solver gives different answers?!!

I'm not a mathematics girl but I'm pretty sure that the variance of a vector X should be a convex quadratic problem. my objective function is as follows: arg min var(sum(L) + X*L) x>0 vector X is ...
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3answers
54 views

Is the derivative of a quadratic related to the second difference of that quadratic?

Please do not judge me too harshly for my lack of knowledge, but at school we have gone over Quadratic functions recently. Now, these types of functions are not new to me, however when we viewed a ...
2
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3answers
42 views

Set of real $a$ so that the inequality is defined but isn't true for a real $x$

$$x(x-\sqrt {4+\log_a7})\lt \log_7 \frac a{49}$$ I reach the interval $(0,1)$ after looking for the discriminant of the quadratic to be less than zero. However, the solution in the book is an ...
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1answer
29 views

Can principal curvatures be complex numbers in a real world situation?

Can the equation for the principal curvatures, $k^2 - 2Hk + K = 0$ (where H is equal to the mean curvature and K is equal to the Gaussian curvature), ever have complex roots? In other words, where ...
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1answer
22 views

A problem with Maple Polynomial Curve Fitting

I've been encountering a problem while trying to use the method of least squares to fit a quadratic polynomial. Below is the question 1) Use the method of least squares to fit a quadratic polynomial ...
6
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2answers
554 views

If p(x) is a non-negative quadratic polynomial, p(0)=8 and p(8)=0, what is p(-4)?

A quadratic polynomial $p(x)$ is such that $p(x)$ never takes any negative values. Also, $p(0)=8$ and $p(8)=0$. What would $p(-4)$ be? I tried doing it by taking the minimum value as zero that is ...
3
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1answer
26 views

Is my answer for this quadratic graph question correct?

There is a question in my maths exercises textbook that I have got a different answer than the one given in my textbook. The question is :- For the following graph of the quadratic equation $$ y = ...
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3answers
41 views

values for which quadratic curve lies below x axis

A quadratic equation $y=(k+1)x^2-3x+(k+1)$ we need to the find the set of values of $k$ for which the curve $y$ lies below the $x-$ axis. I used the quadratic formula and equate it to $0$ $ 3\pm ...
6
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2answers
95 views

Solutions to $(4x^2+\frac{16}3x)^{\sqrt {3-x}}=(4x^2+\frac{16}3x)^{\sqrt {2x+11}-\sqrt{x+2}}$

$$(4x^2+\frac{16}3x)^{\sqrt {3-x}}=(4x^2+\frac{16}3x)^{\sqrt {2x+11}-\sqrt{x+2}}$$ I found the solutions to be $0, -\frac32, -1, -\frac43$ I can't figure out why any of those wouldn't work, but my ...
22
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5answers
3k views

I think I can complete the square of any quadratic, is it true? (Any reason to ever use Quad. Formula?)

I was taught that you could only complete the square of a quadratic if the coefficient on the $x^2$ term is 1. However, playing a little bit with other quadratics, I've found that it's just not ...
3
votes
2answers
74 views

How do you solve this quadratic equation?

The number of values of a for which $$ (a^2 - 3a + 2)x^2 + (a^2-5a + 6)x + a^2-4 = 0$$ is an identity in x is? Here's how much I was able to solve through:- $$ (a^2 - 3a + 2)x^2 + (a^2-5a + 6)x + ...
3
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1answer
50 views

The biggest and smallest integer solution of $\sqrt{(5+2\sqrt6)^{2x}}+\sqrt{(5-2\sqrt6)^{2x}}\le98$ are?

$$\sqrt{(5+2\sqrt6)^{2x}}+\sqrt{(5-2\sqrt6)^{2x}}\le98$$ I noticed that $5+2\sqrt6=(\sqrt2+\sqrt3)^2$ but that hardly helps. After cancelling the square root and the power of two, I tried ...
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6answers
99 views

Find all solutions exactly for $2\sin^2 x - \sin x = 0$

How can I solve this Question ? Find all solutions exactly for $2\sin^2 x - \sin x = 0$ my answer was $( 2k\pi,\pi+2k\pi, 11\pi/6, 7\pi/6)$ but my teacher answer was $( 2k\pi,\pi+2k\pi, \pi/6 ...
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1answer
17 views

Distance and speed question on minimum time

Suppose that there is a narrow bridge 4 metres wide that only 1 bus can pass on it. The bus is travelling at 10m/s and it is moving with constant speed. A man is 2m away from the bus and is crossing ...
0
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2answers
39 views

What does represent this equation?

$y^2 - 3z^2 + 4xz = 4$ Find its axis. As accurately as possible sketch its intersection with the plane $y = 0$. I tried with the making matrix showing the equation.
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0answers
27 views

Taylor expansion need help understanding.

I am at the moment reading a paper (SURF) and trying to understand what is happening here and how the things works as it does.... a non maximum supression is performed on the scale space ...
0
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0answers
14 views

Second-order quadratic model with bias term

I have 3 points in the 3-d space and I would like to estimate the parameters of a second-order quadratic model with a bias term $z=f(x,y)=ax^2+bxy+cy^2+dx+ey=\theta^TQ\theta+\eta^T\theta$ where the ...
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2answers
48 views

Two circular tangents [closed]

The total area of both circles is $230$ $m^2$, i need to find the radius of each circle. The circles are externally tangential and the distance from their centers are $11$$m$. Unable to upload ...
0
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2answers
21 views

If a line parallel to $y=-7x+3$ touches the parabola $2x^2-3x+2$ in the point $(x_0,y_0)$, what is the value of $4x_0+y_0$?

I tried solving this but I've no idea how to find the point where a line of the form $y=-7x+n$ intersects a given parabola. Hints are welcome. Don't know calculus (don't even know if it's applicable ...
3
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4answers
52 views

The sum of the cubes of the reciprocal values of the roots of the equation $x^2+ax+1=0$ is?

The equation : $$x^2+ax+1=0$$ $a$ is a real number. How to find the sum of cubes of the reciprocal of roots for this equation. I tried solving this just by brute forcing it, but I get expressions ...
0
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1answer
33 views

Intersection of two parabolas where one is vertex shifted

I would like to be able to calculate the intersections of two parabola's which accounts for one or both of the parabola's being shifted along the x axis I have written an excel vba function to do ...
1
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1answer
38 views

Steve Nash’s expected value from his one-and-one free throw situation is 1.72 points. What is his free-throw percentage?

The one-on-one free throw situation works like this - for the first throw, if you make it, you get to do it again. If you miss, you don't get another chance. If you make it the second time, you get ...
2
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0answers
150 views

Lagrange multiplier expression

I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1\\ \text{subject to} & ...
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1answer
119 views

Exercise about an algebraic surface

Let $\mathbb{P}^6$ the six-dimensional complex projective space. Suppose that $Q_{i}$ is a smooth quadric in $\mathbb{P}^6$ for $i=1,...,4$. Define $$S=Q_1 \cap Q_2 \cap Q_3 \cap Q_4 $$ as complete ...
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1answer
41 views

Estimation of Quadratic form parameters and convexity/concavity surface

I have 3 points in the 3-d space with their coordinates $(x~y~z)^T$. I would like to find the expression of the $\textbf{concave}$ quadratic surface that form those 3 points, i.e., $z=f(x,y)=ax^2 + ...
0
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2answers
47 views

I have no idea what below surface equation represent

I have the equation: $$x^2+y^2+4z^2-14xy+8xz-8yz=24$$ What does this equation represent? How can I find the "axes" of it (?), and is it possible to draw it when it intersect the plane $z=0$?
2
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1answer
66 views

Solutions of $\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$

$\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$ I get that $-3$ as a solution, but apparently 1 is as well a solution, and I don't see a mechanism through which I could find it. Any help would be ...
0
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1answer
20 views

Values of $p$ for which equation $p3^x+2\cdot 3^{-x}=1$ has a unique solution

$p3^x+2\cdot 3^{-x}=1$ I got this down to a quadratic equation by marking $3^x$ as $t$ and I fiddled with the stuff and got some solutions that apparently don't fit the real one in the textbook was. ...
0
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0answers
53 views

Confusion regarding dF/dx=0, F=constant

I thought I found a theorem Given a curve in the $(y,x)$ plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of $F$ along this curve satisfies relation $g = ...
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2answers
53 views

Dual plot for complex roots of quadratic equation

Real roots of quadratic equation $ x^2 - \sqrt 3 x + 1/2 =0 \tag{1} $ can be plotted on $x$- axis as its parabola intersection at $ (\sqrt 3/2 \pm 1/2,0). $ In an improvization I assign ...
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3answers
76 views

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then $a=b=c=0$ [closed]

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then it is an identity i.e. it is true for all values of $x$ and $a=b=c=0$. What is a proof of this?
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0answers
24 views

Solving the quadratic optimization problem with quadratic inequality constraint

I have a quadratic optimization problem which which both objective function and constraint are convex. As the problem is very big, I used decomposition technique and divide the problem to smaller ones ...
0
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1answer
22 views

Finding Both Missing Co-ordinates in distance formula

Hi I am using this to find location of a device in a 2d plane based on the distance formula. The co-ordinates of reference points and the distance of the device from the device is known. How can we ...
19
votes
5answers
325 views

Probability of $ax^2 + bx + c = 0$ having real solutions

$a$, $b$, $c$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $ax^2 + bx + c = 0$ has real ...
5
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4answers
145 views

How to solve an equation with $x^4$?

Today, I had this question on a Maths test about Algebra. This was the equation I had to solve: $$(1-x)(x-5)^3=x-1$$ I worked away the brackets and subtracted $x-1$ from both sides and was left with ...
3
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2answers
55 views

Can anyone help me solve this?

Two taps A and B can fill a swimming pool in $3$ hours. If turned on alone, it takes tap A $5$ hours less than tap B to fill the same pool. How many hours does it take tap A to fill the pool? ...
0
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1answer
59 views

Reverse Polish Notation Quadratic formula

The quadratic formula is $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ I tried converting this to RPN; I am new to doing this, and I have thus: b-ac*4*-b2^+±a2*/. Am I ...
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1answer
40 views

Real world example of need for quadratic equation

I am (re)learning the quadratic equation. Having a concrete understanding of its purpose would really help, but I can not find any examples of a real-world scenario that requires the use of it that ...
3
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4answers
211 views

Solving equations with exponentials and a non-exponential term.

I know how to solve exponential equations. Just use logarithms, e.g., $$ 2^x-3=0 \\ 2^x=3 \\ x=log_23 \\ $$ But on a recent math test I found an equation of the form: $$ 2^{n-3}=\frac {20}{n} $$ ...
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2answers
54 views

To find $x$ in $x^2 -8x-11=0$ [closed]

$x^2 -8x-11=0$ I have tried factorising but it won't factorise into a quadratic equation Hi, It would be great if you could complete this question with working and post it. Thx The two solutions of ...
4
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3answers
196 views

How to find cotangent?

Need to find a $3\cot(x+y)$ if $\tan(x)$ and $\tan(y)$ are the solutions of $x^2-3\sqrt{5}\,x +2 = 0$. I tried to solve this and got $3\sqrt{5}\cdot1/2$, but the answer is $-\sqrt{5}/5$
5
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3answers
57 views

Intuitive understanding of factoring quadratic equations [duplicate]

When factoring a second degree equation $ax^2 + bx + c$ you find the roots then take $a(x - \text{root})(x - \text{root})$. I am wondering why this works. Sorry if poorly phrased question.
2
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2answers
53 views

Finding conditions to make roots of a quadratic less than one in magnitude

I'm doing a problem that asks for you to find the conditions that make $y$ defined: $$y=x^2-bx+c$$ have real roots with magnitude less than one. Now the condition for the roots being real seems to ...
0
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5answers
50 views

How to know if equation can be solved by factorising before trying?

So, I have core 1 test tomorrow and there is a lot of solving of quadratic equations without calculator and my weakest point is the time I waste in trying to factorise and equation but then it ends up ...
3
votes
3answers
37 views

Finding real coefficients of equation given that $a+ib$ is a root

Below is the question present in a past examination paper. I'll be giving my attempts and how I thought it through. Do feel free to point out any mistakes I make throughout my working even if ...
1
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2answers
54 views

Let $y=x^2+ax+b$ cuts the coordinate axes at three distinct points. Show that the circle passing through these 3 points also passes through $(0,1)$.

Let $y=x^2+ax+b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$. Since, the graph of the ...
3
votes
4answers
78 views

Number of fingers of a Martian

I have a question about what seems to be modular arithmetic, but I can't quite get the answer. The problem goes along the lines of: It is often said Earthlings use the decimal system because they ...