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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36 views

For a natural number b, N(b)= number of natural numbers a such that the equation x2+ax+b=0 has integral roots.

For a natural number $b$, $N(b)= $ number of natural numbers $a$ such that the equation $x^2+ax+b=0$ has integral roots. What is the lowest possible value of $N(6)$?
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1answer
36 views

Integer solutions of a quadratic equation with combined variables

I'm having problems with finding all possible integers solutions of particular equations, like this one for example: $x^2 -xy + 2y^2 = 29$. What sets me off, is the term $xy$, I don't know how to deal ...
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1answer
41 views

how to solve this : $R(m)=12+5m -0.4m^2$? [closed]

How to solve this : $R(m)=12+5m-0.4m^2$? Jeremy's junkers keeps track of the revenue it makes from selling used exhaust manifolds. they have found that the revenue is a function of the number sold ...
1
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2answers
302 views

Solve $y^2 + 3xy - 10x^2 + y + 5x = 0$ for y in terms of x

I'm given the following equation: $y^2 + 3xy - 10x^2 + y + 5x = 0$ and asked to solve $y$ in terms of $y$. My attempt: $y^2 + (3x+1)\times y - 10x^2 + 5x = 0$ $\Rightarrow (y+(3x+1)/2)^2 - ...
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3answers
87 views

how do you solve $(x^2-5x+5)^{x^2-36} =1$

Can someone please show me how they would work it out as I have never come across this before. $$(x^2-5x+5)^{x^2-36} =1$$
2
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1answer
37 views

What's so special about quadratic extensions?

Reading through chapter 13 "Field Theory" from Dummit and Foote Algebra. I am wondering why such an emphasis is placed upon "quadratic extensions" of a field F. They state that for any field F ...
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3answers
137 views

Solving a rational equation with multiple and nested fractions

This is the equation to solve: $\dfrac{\dfrac{x+\dfrac{1}{2}} {\dfrac{1}{2}+\dfrac{x}{3}}}{\dfrac{1}{4}+\dfrac{x}{5}}=3$ What I did: $x+\dfrac{1}{2}=\dfrac{2x+1}{2}$ ...
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5answers
68 views

Better way of solving this quadratic equation?

This is the excersice: $$x^2 - 2mx - 2m - 1 = 0$$ I've done this: 1.- $$x^2 - (2mx -2m) - 1 = 0$$ 2.- $$x^2 - (2m(x-1)) - 1 = 0$$ I think I need to transform: $$(2m(x-1))$$ to something of the ...
1
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4answers
79 views

Is this the correct formula for this quadratic equation?

I'm doing some excersises, but i'm not sure who to apply the 'formula' given: $$x^2 - 2px + p^2 - 1 = 0$$. I've found this formula on my book: Is it the correct 'formula'? If it were something ...
2
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2answers
76 views

Solve in $\mathbb{Q}$ the equation $x^2-(\sqrt{2}+1)x+\sqrt{2}=0$.

Solve in $\mathbb{Q}$ the equation $x^2-(\sqrt{2}+1)x+\sqrt{2}=0$ Somebody can help me? I dont remember how to do.
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1answer
53 views

Find the sign of $a,b,c$ in $ax^2+bx+c$ given the graph and a coordinate on it.

So my first approach was that, we see that there are $2$ roots. And one is negative and one is positive. $a$ would be evidently positive. The positive one's modulus is bigger than the negative ...
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3answers
61 views

Completing the square help

The textbook gives this equation: ${12x^2 + 24x -8x = 0}$ with an answer of ${x = 0}$ or ${x = -{4\over3}}$ But I suspect it should be ${12x^2 + 24x -8 = 0}$ So in order to solve this, I would ...
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2answers
72 views

Soft Question: Weblinks to pages with explanation on quadratics.

I recently placed a question based on quadratics and received a few valuable answers. One of them was a comment in an answer with a link in it which I found useful. But unfortunately the webpage (of ...
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2answers
52 views

Finding $a$ yielding minimum value for quadratic root expression $(x_1+2x_2)(x_2+2x_1)$

The problem is: We have the expression $(x_1+2x_2)(x_2+2x_1)$, where $x_1$ and $x_2$ are the roots of $$f(x)=x^2+ax+a+\frac{1}{5}$$ Find the value(s) of $a$ yielding the least possible value for ...
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1answer
27 views

Hilbert Symbol over $\mathbb{R}$ (bilinearity)

Let $\mathbb{R}$ be the field of the reals and let $a,b,c \in \mathbb{R}^{\times}$. As you probably know, the Hilbert symbol over any field $K$ is defined as: $$(\frac{a,b}{K}) = 1 \text{ if } \exists ...
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5answers
155 views

Finding range of $m$ in $x^2+mx+6$.

Find the range of values of $m$ in the quadratic equation $x^2+mx+6=0$ such that both the roots of the equation $\alpha,\beta<1$. My attempt - it is given that $\alpha<1$ and $\beta<1$ ...
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1answer
53 views

Stuck while seeking $a$ with which the difference between quadratic roots is above 3

The problem is: Find the values of a with which the roots of the following inequality will form an interval longer than 3: $$x^2-(a^2+3a+1)x+a^2+3a^3\le0$$ From Vieta's formulas, the ...
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1answer
23 views

How get the standard form just with x-intersects and vertex coordinates?

I was trying to get the standard form to an equation where: The graph intersects the x-axis at x=−1 and x=3; The vertex of the graph lies at (1,2); I can't figure it out even if I got to resolve ...
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3answers
36 views

quadratic equation with zero product

I have this equation ${3x^2 -12 = -9}$ The answer the text book gives is ${x = 1}$ or ${x = 3}$. But I would solve it by first of all dividing 3 on both sides which gives: ${x^2 -4 = -3}$ Then ...
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2answers
48 views

Finding values of $a$ with which two equations are equivalent; getting rid of radical sign

Two equations are given: $$x^2+(a^2-5a+6)x=0$$ $$x^2+2(a-3)x+a^2-7a+12=0$$ We need to find the values of $a$ that will render them equivalent. From the first equation, $$x=-a^2+5a-6$$ From the ...
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1answer
34 views

Solving a exponential quadratic equation

Find t given: $$0.715+258.115e^{-0.5t}-67.83e^{-0.25t}=0$$ Please help. I've substituted $e^{-0.5t}$ for $x$ and did $$-67.83x^2+258.115x+0.715=0$$ but the answer is not coming out? Am I doing ...
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6answers
893 views

If $3x^2 -2x+7=0$ then $(x-\frac{1}{3})^2 =$?

If $3x^2-2x+7=0$ then $$\left(x-\frac{1}{3}\right)^2 =\text{?}$$ I am so confused. It is a self taught algebra book. The answer is $\large -\frac{20}{9}$ but I don't know how it was derived. ...
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4answers
116 views

The behavior of quadratic formula in the limit $a\to 0$

$$\lim\limits_{a\to0}\frac{-b+\sqrt{b^2-4ac}}{2a}$$ $b$ and $c$ are constants. As $a$ approaches $0$, what does the formula approach? Example: ...
2
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1answer
93 views

how do you solve $(x^2-5x+5)^{x^2-36}=1$? [closed]

The question is: Solve $$(x^2-5x+5)^{x^2-36}=1.$$ Many Thanks
3
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5answers
57 views

Quadratics question

To solve $-3x^2 +2x +1=0$, I'd normally break the middle term and then factorise. But I was wondering if there was a way to skip the factorising step? The factors I'd use in place of the middle term ...
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2answers
42 views

quadratic reduction problem

A train is travelling between two stations that are $100$ km apart at a speed of $v$ km/h. Express the time taken for the journey in terms of $v$. Here I got $\ t=\dfrac{100}{v}$. On the return ...
3
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1answer
82 views

Show that there are numbers c and d such that F(A) = cTr(A^2) + d(Tr(A))^2,

Suppose F(A) is a quadratic function of a real symmetric matrix, A. This means that there are numbers $f_{ijkl}$ so that F(A) = $\sum_{ijkl}f_{ijkl}a_{ij}a_{kl}$. Suppose that $F(A) = F(QAQ^t)$ for ...
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2answers
91 views

Why am I getting two answers for 8th root of continued fraction

Find value of $x$: $x=\sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-....and\,so\, on}}}$ On solving ,we have $x^8=2207-\frac{1}{x^8}$ $x^8+\frac{1}{x^8}=2207$ $x^4+\frac{1}{x^4}=47$ ...
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1answer
53 views

a very basic question on finding the discriminant for $x^2+2(a-3)x-3a-7=0$

Sorry for asking such a basic question. In the following quadratic equation $$x^2+2(a-3)x-3a-7=0$$ by my calculations, $$D=\left(\frac{b}{2}\right)^2-ac=(a-3)^2-1(-3a-7)=a^2-6a+9+3a+7=a^2-3a+16$$ ...
5
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1answer
150 views

Prove that a sequence whose second difference is a nonzero constant is quadratic.

For example, if {$a_0, a_1, a_2, a_3, ...$} is the sequence, the first difference is {$a_1-a_0, a_2-a_1, a_3-a_2, ...$}, and the second difference is {$(a_2-a_1)-(a_1-a_0), (a_3-a_2)-(a_2-a_1), ...
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2answers
39 views

Find the sum of cubes of roots of a biquadratic

Given that $a,b,c,d$ are the roots of the equation $x^4-3x^3+x^2-2x+1=0$, find the value of $a^3+b^3+c^3+d^3$. Since, there are no 'zero' coefficients in the equation, it looks like a tough job for ...
1
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2answers
76 views

Confused about a missing $\pm$ sign in texbook's answer (simple quadratic equation)

The problem goes like this: Working together, two cranes unload a barge in $t$ hours. What time does it take for each crane to unload the same barge on its own, provided that crane 1 spends $a$ ...
2
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5answers
74 views

What will change if we admit a different definition of $\sqrt a$

We know that $\sqrt a$ is the non negative solution of the equation $x^2=a$ with $a\geq 0$. So if we want to solve the equation $x^2=a$, we say that $x=\pm\sqrt a$. How will mathematics be affected ...
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2answers
87 views

Completing the square of $(x+a)(x+b)$

The problem is simple, to complete the square of $(x+a)(x+b)$. My calculations yield $$\left(x+\frac{a+b}{2}\right)^2-\frac{(a+b)^2}{4}+ab,$$ But the textbook's answer is different ("problem 361", ...
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2answers
91 views

Need help with an Elementary Math question [closed]

If $a+b+c=1$ and $ax^2 + bx + c = 0$ has a unique solution. Find $a,b$, and $c$.
3
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1answer
69 views

How to quantify how much my data resembles a linear relationship?

I have a bunch of data points in Excel for different test subjects that are each represented by unique colors in the following graph: I wish to quantify in Excel what seems obvious to my eyes, that ...
6
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6answers
734 views

Quadratic formula - check my simplificaiton

I am trying to solve this equation using the quadratic formula: $$x^2 + 4x -1 = 0$$ I start by substituting the values into the quadratic formula: $$x = {-(4) \pm \sqrt {(4)^2 - 4(1)(-1)} \over ...
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4answers
625 views

Exponential Simultaneous Equations

Solve the following simultaneous equations: $$2^x + 2^y = 10$$ $$x + y = 4$$ Looking at it, it is obvious that the answers are $(3,1)$ and $(1,3)$, however, I was wondering if they could be solved ...
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2answers
28 views

Getting the quadratic function given the vertex and one point.

Find out the quadratic function for the parable that contains the point $(1,1)$ and the vertex $(-2,3)$. The notes I got are pretty vague: $$b = 4\frac{-2}{9} = \frac{-8}{9}\\ c = ...
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2answers
43 views

an unclear step in a textbook solution of quadratic inequality

We have a quadratic inequality $$Ax^2+Bx+C>0$$ After solving it for cases where $B^2-4AC > 0$, my textbook turns to cases where $B^2-4AC < 0$: Using the perfect square method, let's ...
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1answer
28 views

Square roots of quadratic functions

Consider the real-valued function of a real variable $f(x) = \sqrt {ax^2 + bx + c}$ with $a$, $b$ and $c$ given and $a>0$. When $\Delta = 0$, the function is equal to $|\sqrt{a} ...
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1answer
42 views

If 2 roots of the equation $(p-1)(x^2+x+1)^2-(p+1)(x^4+x^2+1)$ are real and distinct and $f(x)=\frac{1-x}{1+x}$…

Question: If 2 roots of the equation $(p-1)(x^2+x+1)^2-(p+1)(x^4+x^2+1)$ are real and distinct and $f(x)=\frac{1-x}{1+x}$, then $f(f(x))+f(f(\frac{1}{x})) = ?$ (a)p (b)2p (c)-p (d)-2p Attempt: ...
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3answers
132 views

What is equating the coefficient of the corresponding power of x?

I recently asked a question on mathematics.SE pertaining to solving an unusual quadratic [1], and was introduced to the phrase 'equating the coefficient of the corresponding power of x'. What does ...
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0answers
59 views

Minimization of a multivariate quadratic equation

I am interested in the minimum of a general multivariate quadratic equation for non-negative real numbers: $$ \begin{aligned} & \underset{x_i}{\text{minimize}} & & ...
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3answers
123 views

How to solve the equation $x^2 - 6x + 25 = 0$

There is one thing that I don't understand. How is it possible that $-6x$ can be into "$9$". Can you describe it to me by calculation? Thanks!
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0answers
23 views

Reverse Taylor series for sine

I want a little help with reverse Taylor series for sinus if is possible :D .From what I read the formula is: RadOfAngle - RadOfAngle^3*3! + RadOfAngle^5*5! - RadOfAngle^7*7! = Sins value. How can I ...
4
votes
2answers
80 views

How do I prove that the $f(x)$ is positive for all real $x$?

$$ \frac {f(x+y) - f(x)}{2}= \frac{f(y)-a}{2} +xy $$ for all real $x$ and $y$. If $f(x)$ is differentiable and $f'(0)$ exists for all real permisible values of $a$ and is equal to $\sqrt{5a-1-a^2}$. ...
2
votes
5answers
80 views

Quadratic formula does not work

If I put the equation: $5x^2-x-4 =0$ in the quadratic formula, than I get $x = 1$ or $x = \frac{-4}{5}$ but the real zeros are: $x = -1$ or $x = \frac{4}{5}$ Can somebody explain me if the ...
2
votes
2answers
74 views

For what values of $ a, b$ does the equation have real roots?

For what values of $a,b$ does the equation $${ x }^{ 2 }+2\left( 1+a \right) x+\left( 3{ a }^{ 2 }+4ab+4{ b }^{ 2 }+2 \right) = 0$$ have real roots? For it to have real roots, the ...
0
votes
1answer
51 views

Solving algebraic equations with radicals

I have several problems requiring assistance. Solve for x: $x\left( x-\sqrt { 3 } \right) \left( x+1 \right) +3-\sqrt { 3 } \quad =\quad 0$ I've followed the suggestion to get x^2 - (√3 -1)x + ...