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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1answer
116 views

Dimensions of a paddock (3 sides of a rectangle) to enclose maximum possible area

I need help with Qs 4, 5 and 6!! Three sides of a rectangular paddock are to be fenced, the fourth side being an existing straight water drain. If 1000m of fencing is available, what dimensions ...
2
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4answers
74 views

Simultaneous Quadratic Equations: $x^2 + y ^ 2 - 2 x + 6y - 35 = 0$ and $2x + 3y = 5$

I've been given the task to simultaneously solve: $$x^2 + y ^ 2 - 2 x + 6y - 35 = 0$$ $$2x + 3y = 5$$ I've tried applying the substitution method by reordering the second equation to both $x$ and ...
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2answers
12 views

Finding the set of all $k\in R$ so that the roots of equation $x^2-(5k+3)x+(k+3)^2$ obey $x_1<4<x_2$

Stuck on this one. I get inequations with square roots in them, ones which we aren't supposed to know to solve this problem.
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2answers
24 views

The equation $x^2+px-p$ has two real and different solutions $x_1$ and $x_2$ for which $1<(x_1/x_2+x_2/x_1)<3$ for which interval of $p$?

This is a multiple choice question, but I figure it will be easy enough to do without the given answers. It seems like an easy question. Use Vieta's formula from the inequality and define $p$ from ...
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1answer
40 views

The first assumption leads to the third one that looks inconsistent at a glance. Can you explain it better?

Background I am trying to solve the following problem: > Given 2 distinct curves $C_1: y=f(x)=e^{6x}$ and $C_2: y=g(x)=ax^2$ where $a>0$. The objective is to find the range of $a$ such that ...
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0answers
50 views

Line intersection with Sphere

I'm trying to get a formula for calculating intersection points of a line with a sphere (3d space). I've been following this one: Wiki Line-sphere intersection But I'm 99% sure that this one is ...
3
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2answers
178 views

Simple maths and a typo: impossible answer?

Long story short: While doing some simple math exercises, I came across one that seemed impossible. Days later I decided to search the web for it, and found out there was a typo, putting the $^2$ ...
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1answer
29 views

Difficulty turning a quadratic equation to “vertex”-form

I'm having difficulty reducing a quadratic equation to its "vertex-form" by following my textbook and nearly every tutorial I can find online. The starting equation is: $$f(x) = -2x^2 + 16x - 24$$ ...
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2answers
71 views

Online service for completing the square

Is there any online service for completing the square? For example:
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6answers
74 views

If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then find value of m.

Problem : If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then find value of $m$. My Approach: $|x^2+4x+3|-mx+2m=0$ Case I : $x^2+4x+3-mx+2m=0$ $\Rightarrow x^2+ x (4-m) + ...
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0answers
37 views

The property of roots of quadratic equation

I have a problem with two tasks: Given a quadratic equation $ax ^ 2 + bx + c = 0 $ (roots can be complex or real), $a, b, c \in Q$. Prove that ${x_1} ^ m + {x_2} ^ n \in Q$. We have a trinomial ...
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4answers
56 views

How can I factorize this quadratic expression

Going by the exercises of a book I have been factorizing quadratic equations the following way, let's say I have: $$ {x^2 - 7x + 12 = 0} $$ I know that $$ {a \times b = 12 \\ \text{ and } \\ a + b ...
6
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2answers
355 views

quadratic equation what am I doing wrong?

solve $$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} $$ $$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} \Rightarrow $$ $$ 5x+19 = (x+7) + 4\sqrt{x-5}\sqrt{x+7} + (x+5) \Rightarrow $$ $$ 3x + 17 = ...
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1answer
50 views

Solving a quadratic involving square root

$$\sqrt{\frac{x}2} = 1-x$$ so $$x = ?$$ I have tried to solve many times and i got $x = \frac52$ everytime. But my book says answer is $\frac12$. I think i couldn't understand square roots clearly.. ...
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4answers
38 views

Finding value of $m$ which is a part of a quadratic

Q : $$m \gt 2$$ $$x^2 + (m-3)x - 2 = 0$$ If $|x_1 - x_2| = 3$, so $m = ?$ Stuck here. Please give me a hint
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3answers
260 views

Solving an operation involving roots of a quadratic equation

A question from my book: $3x^2 + 7x + 5 = 0$ So, $\sqrt{(x_1^2 + 2x_1x_2 + x_2^2)} + x_1x_2 = ?$ Options: A) $4$ B) $5$ C) $6$ D) $7$ E) $8$ It's looking too easy, my answer is $-\frac{2}3$, but it ...
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2answers
64 views

An equation with negative exponents in quadratic equations test

There is a problem like this : $x^{-1} = 2x^{(-1/2)} + 3 , x = $? in my test. I'm working on it for a half of hour but still i can't solve. Please help me. (Excuse my bad grammar)
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1answer
64 views

Finding equation of parabola when given point and tangent line

I have tried substituting the coordinate points (2,4) and the formula for the turning point (-b/2a, etc.) into the equation and solving it simultaneously, however this has not seemed to work. The ...
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3answers
54 views

$ax^2+bx+c=0$ has roots $x_1,x_2$. what are the roots of $cx^2+bx+a=0$.

Given solution: Dividing the first equation by $x^2$ we get $c(\frac{1}{x^2})+b(\frac{1}{x})+a=0$ so $(\frac{1}{x_1}),(\frac{1}{x_2})$ are the roots of $cx^2+bx+a=0$.{How?It is not obvious to me.} ...
2
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2answers
56 views

Help me to prove this statement about quadratic equations? (from Gelfand's Algebra).

$ x^2+px+q=0 ${p,q are integers; a,b are roots}. Prove $a^n+b^n$(n is any natural number) is an integer. This is the third part of the problem.I have previously proved that $a^2+b^2$ and $a^3+b^3$ ...
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2answers
66 views

Difference between two real roots with uniformly distributed coefficents

I have a question that first I need to know what is happening, but then I also need to code it in a program called APPL, which is an extension from Maple18 that I really have never used, yet I have ...
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2answers
65 views

calculate the intersection of two number series

I have a series of numbers. It is in the form of a parabola. This series is guaranteed to have at least one perfect square within it (edited I thought there was only one). The second series is also a ...
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3answers
34 views

Relationship between roots and equations

I'm stuck on topic of relationship between roots and equations. The roots of $x^2 -2x +3 =0$, are $\alpha$ and $\beta$. Find the equation whose roots are : 1- $\alpha+2$, $\beta+2$ 2- $\alpha^2$, ...
7
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4answers
309 views

Why are there four solutions to $x^2-2x-8=0$ in $\mathbb{R}$? Or am I wrong?

It might be a very trivial question to ask but why do we get four different solutions for a quadratic equation using these two methods? $x^2-2x-8=0$ We see that factors are $(x-4)$ and $(x+2)$ so ...
2
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2answers
50 views

How do I complete the square when the $x^2$ has a coefficient greater than $1$?

For homework we are doing completing the square and a few of them have coefficients greater than one. For example one of the quadratic equations we have to complete the square of is $-2x^2-7x-2$. All ...
0
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1answer
31 views

Complex Roots Of a equation - Equilateral triangle

$z_1$ and $z_2$ are the roots of $3z^2+3z+b=0$.If $O(0),A(z_1),B(z_2)$ is an equilateral triangle then what will be the value of b ? My approach:I took $z_1=m_1+in_1$ and $z_2=m_2+in_2$ and proceeded ...
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0answers
22 views

Alternative method for y-vertex calculation

So, I've been wondering the following: If you can determine the x coordinate of the vertex of a quadratic function by averaging the x coordinates of both roots, would it be possible to determine the ...
0
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1answer
43 views

Derive Equation from the set of values

I have a set of values $$\begin{array}{|c|c|} \hline\text{$X$} & \text{$Y$} \\ \hline 1 & 2 \\ \hline 2 & 10 \\ \hline 3 & 30 \\ \hline 4 & 68 \\ \hline \end{array}$$ ...
1
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1answer
185 views

Show that quadratic is positive for all real values of x

I have been asked this question: Show that $x^2 + 2px + 2p^2$ is positive for all real values of $x$. I've worked it out like so: Discriminant = $(2p)^2 - (4\times 1\times(2p^2)) = 4p^2 - 8p^2$ I ...
1
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2answers
32 views

Figuring domain of constant $a$ in a equation with some condition

Here is what questions says Question: If $a\in \mathrm{R}$ and the equation $-3(x-[x])^2+2(x-[x])+a^2=0$ (where $[\cdot]$ denotes the greatest integer $\leq x$) has no integral solutions, then all ...
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1answer
38 views

Quadratic Functions Word Problem Help

Two numbers differ by 18. determine the two numbers if the sum of their square is 3860.
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1answer
25 views

Why is this answer wrong? (point of intersection between parabola and line)

Question: Use the discriminant to determine the number of points of intersection of the line $y=3x+5$ and the quadratic functions $f(x)=3x^2-2x-4$. Solve to find the points of intersection. ...
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3answers
34 views

Find the speed of a jet given the time of travel back and forth

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
0
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2answers
41 views

Converting from factored to standard form: why is this answer wrong?

Converting the equation $$y=-2(x-2+\sqrt{5})(x-2-\sqrt{5})$$ to standard form seems to give $$-2x^{2\space }+3.528x+6.4171392.$$ My handout tells me that the answer is different. What is wrong here? ...
2
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1answer
158 views

Why is this answer wrong? (quadratic functions)

Question: Determine the quadratic function that has the given roots (x-intercepts) and passes though the given point. $x=2+\sqrt{5}, x=2 - \sqrt{5}$. The given point is $(2,10)$. This is my ...
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0answers
30 views

How do I solve this quadratics problem? [duplicate]

The problem: A jet flew from Tokyo to Bangkok, a distance of 4800km. On the return trip, the speed was decreased by 200 km/h. If the difference in the times of the flights was 2 hours, what ...
2
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4answers
67 views

How to factor quadratics $(x^2 + 4x + (-357) = 0)$

I need to find $2$ factors of $-357$, which add up to $4$. Obviously one number is positive and the other is negative. I understand this and I know the factors can be $21$ and $-17$; but, how do I ...
0
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1answer
42 views

Which are the conditions for a biquadratic equation to have 4 different roots?

Which are the conditions for a biquadratic equation to have 4 different roots in R? I think D>0, If we have $$t=x^2$$ then t>0. Is there any other condition?
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0answers
69 views

Minimum Curvature Path

Let's say we are given a closed race track with a given and constant width. I am to implement an algorithm which finds both shortest path trajectory and minimum curvature trajectory for the car. I ...
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1answer
56 views

Find Minimizer and Minimum Value for a Function

I am trying to work through some problems to find the minimizer and minimum value of a function. The book I am using doesn't have a clear cut example and I can't seem to find a good example online ...
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2answers
26 views

fitting a quadratic to 3 coordinates

say I have 3 points on the plane (Cartesian coordinate system), (a,b), (c,d) and (e,f), I am fairly certain that there is one unique quadratic curve which passes through each point. what is the ...
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1answer
17 views

Finding the positive component of a constant in a quadratic equation.

Can you help me to do this question:it is from a past cambridge exam paper Find the positive constants $a$ and $b$ such that $x^4+9/x^4 =[x^2-a/x^2 ]^2+b$ for all non-zero values of $x$. Hence write ...
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0answers
15 views

Determine the multi-dimensional relationship given the data

I have a dependent variable - A and 3 independent variables, H,V and N I have a data for all the variables and dependency relationship is based on my operational knowledge. I'd like to know what ...
0
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3answers
62 views

How to solve this quadratic equation?

So I've got this quadratic equation and am totally unable to solve it. Can someone tell me how to do it? $$\frac{a}{ax-1} + \frac{b}{bx-1} = a + b,$$ where $x$ is not equal to $\frac{1}{a}$ or ...
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3answers
51 views

Quadratic equation involving floor function.

If equations $x^2-3x+4=0$ and $ 4x^2-2\lfloor3a+b\rfloor x+b=0\space (a,b\space\epsilon\space R) $ have a common root then the complete set of values of $a$ is ? I have not yet been able to develop ...
0
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4answers
45 views

If $a,b$ are the roots of the equation $2 x^2 -3 x +1 = 0$, find an equation whose roots are $a/(2b +3)$, $b/(2a +3)$

If $a,b$ are the roots of the equation $2 x^2 -3 x +1 = 0$, find an equation whose roots are $a/(2b +3)$, $b/(2a +3)$ I was practicing quadratic equation questions online but I am stuck on this ...
0
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3answers
113 views

continued fraction of the roots of $x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$

I would like to find the continued fraction expansion of the roots of: $$x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$$ Eq 1.6 from [1] What makes this problem so ...
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0answers
17 views

Constrained Motion Study

I'm working on a motion study for a disk moving within a mechanical enclosure and I'm having trouble reducing my equations. The system can be defined as 4 circles which are bound inside each other. ...
0
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1answer
33 views

Condition for roots of the equation to be real.

Show that for $ 3 > y_1 >0 $ the roots of the equation $$(y_1-2)x^2-(8-2y_1)x-(8-3y_1)=0$$ are real, where $y_1$ is a constant. Due to my difficulties in doing this I would be grateful for ...
0
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0answers
40 views

Proof of the Quadratic formula [duplicate]

Prove: $ax^2+bx+c=0 \implies x=(−b±\sqrt{b^2−4ac})/2a$ I know it's probably simple just can't get my head around it?