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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1answer
25 views

Trigonometric Equation, quadratic using two functions

I am struggling to know how to solve this equation as it involves more than one type of trigonometric function, I know how to do it with one repeated function. If a solution could be explained, that ...
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2answers
20 views

Quadratic using the roots of unity, where $\omega^7 = 1, \omega \neq 1$

Say that $\omega$ is a complex number, where $\omega^7 = 1, \omega \neq 1$. Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. $\alpha$ and $\beta$ are roots ...
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3answers
39 views

When can I divide both sides of an equation if one side is zero

Where K is some positive Integer For the following examples: $$ K(a+b)(p+q)=0 $$ $$ Ka^2+Kbx+Kc=0 $$ Can I just divide both sides of the equation by K (dividing into 0 on the right) and effectively ...
2
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0answers
44 views

Does $2^2=4$ imply $2=\pm \sqrt{4}$?

I read the square root property from the book, College Algebra by Raymond A Barnett and Micheal R Ziegler that, The square root property says, If $A^2=C$ then $A=\pm \sqrt{C}$ I took the ...
4
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4answers
55 views

Find all integers n such that the quadratic $5x^2 + nx – 13$ can be expressed as the product of two linear factors with integer coefficients.

I am unsure of how to approach this problem. I have thought about using the Rational root theorem, but I am unsure if this answers the question being asked. Using the theorem, I get $\frac{p}{q} = ...
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1answer
25 views

Interval in which roots lie.

For a quadratic equation, we have several conditions from which we can determine the interval in which the roots lie. eg: If exactly one of the roots of a general quadratic equation lies in the ...
1
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1answer
23 views

Quadratic function with positive integral coefficients problem

Here is the problem statement: Let $f(x)$ is a quadaratic expression with positive integral coefficients such that for every $\alpha, \beta\; \epsilon\; \Re$, $\beta>\alpha$, $\int_\alpha^\beta ...
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2answers
31 views

Roots of quadratic equation

If the roots of $ax^2+bx+c$ are $\alpha$ and $\beta$, express $\frac1\alpha-\frac1\beta$ in terms of $a$, $b$ and $c$. I know how to express $\alpha+\beta$ or $\alpha\beta$ which is usually enough, ...
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1answer
51 views

dealing with quadratic equations [on hold]

You kick a ball into the air. The ball has an initial velocity of $31 ft/sec$ and is $2 ft$ above the ground when your foot makes contact with it. The equation where $h$ is the height of the ball ...
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2answers
25 views

For which $m$ does the equation $\lvert x^2+5x+4 \rvert +x-m=0$ have more than two real solutions?

When dealing with problems containing quadratic equations, I've never come across one that deals with a number of roots that is bigger than 2. I've tried breaking up the function into two parts. ...
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4answers
49 views

Find the sum of the roots of a quadratic function given the vertex of its graph

Question: At this parabola $$y = ax^2 + bx - c$$ and vertex is $T(3,9)$. What is the sum of roots of this parabola ? Help or give a hint. Thanks
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1answer
28 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 ...
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0answers
31 views

Question about alpha and beta [closed]

If $\alpha$ and $\beta$ are zeroes of polynomial $4x^2-3x+8$. find the value of $\alpha^2-\beta^2$.
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4answers
36 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
11 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
23 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
35 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 ...
0
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0answers
17 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
157 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
25 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
67 views

Online service for completing the square

Is there any online service for completing the square? For example:
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6answers
63 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
33 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
48 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 ...
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2answers
319 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
47 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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2answers
50 views

Show that $b^2=9a^2+4ac$ given that the square roots differ by 3 for a quadratic equation $ax^2+bx-c=0$ [on hold]

If the square roots of $ax^2+bx-c=0$ differ by 3, show $b^2=9a^2+4ac$. How do I show this, from the knowledge that I have about alpha and beta, I can't get it right.
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4answers
36 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
240 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
49 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
20 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
52 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
51 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
61 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 ...
0
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2answers
60 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
29 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- $\alpha^2$, ...
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4answers
296 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
48 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 ...
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1answer
23 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
10 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
28 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
34 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 ...
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2answers
31 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
29 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
18 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
31 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
35 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
149 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 ...
0
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0answers
29 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
votes
4answers
64 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 ...