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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3
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5answers
55 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 ...
-1
votes
0answers
18 views

There always exist $n$,such $m$ is a quadratic nonresidue $\mod n$ [closed]

For any postive integer $m$(non-square),show that: There always exist $n$,such $m$ is a quadratic nonresidue $\mod n$
1
vote
2answers
35 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
votes
1answer
78 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 ...
3
votes
2answers
79 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 $x^8=2207-\frac{1}{x^8}$ $x^8+\frac{1}{x^8}=2207$ $x^4+\frac{1}{x^4}=47$ ...
1
vote
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
votes
1answer
56 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), ...
0
votes
2answers
32 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
vote
2answers
75 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
votes
5answers
73 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 ...
4
votes
2answers
81 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", ...
-1
votes
2answers
88 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
votes
1answer
38 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 ...
7
votes
6answers
662 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 ...
7
votes
4answers
460 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 ...
1
vote
2answers
25 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 = ...
0
votes
2answers
42 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 ...
0
votes
1answer
26 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} ...
0
votes
1answer
37 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: ...
1
vote
3answers
37 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 ...
1
vote
0answers
40 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}} & & ...
1
vote
3answers
120 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!
0
votes
0answers
22 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 ...
3
votes
2answers
66 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
75 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
67 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
47 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 + ...
0
votes
0answers
20 views

Any way to factor, collect variable from this equation?

For a sum of quadratic solutions, is there any possible way to factor out the variable $P$ from the following real function? $QT$ is also a variable, and If it matters, $P > 0$ and all indexed ...
1
vote
1answer
40 views

Strange behavior with coordinate transformation of square and quadrilateral

I am trying to map coordinates from a quadrilateral to a square. The coordinates are square: $(500,900)(599,900)(599,999)(500,999)$ quad: $(454,945)(558,951)(598,999)(499,999)$ where the $i^{th}$ ...
0
votes
3answers
61 views

If $ax^2+bx+c=0,a\neq 0$ has two distinct roots,then $a(5a+2b+c)$ is

If $ax^2+bx+c=0,a\neq 0,a,b,c\in \mathbb{R}$ has two distinct real roots in $(1,2)$,then $a(5a+2b+c)$ is $(A)$positive $\hspace{1cm}(B)$negative $\hspace{1 cm} (C)$zero $\hspace{1 cm}(D)$none of ...
0
votes
2answers
28 views

Solve the algebraic expression for a, b, and c of the function x

I am trying to solve for $a$, $b$, and $c$ in the expression below, but I have found that the way I tried to solve it is convoluted and did not work out. I believed that by solving for x, I would be ...
0
votes
1answer
24 views

how to solve the question with one unknown variable?

This is the Question: 5 years ago, Ebo was 3 times as Old as Atu.In 3 years, Ebo will be twice as old as Atu. What is the sum of their ages now? I am easily able to solve it with two unknown ...
-1
votes
0answers
24 views

analytical solution of non-linear least square problem

I am implementing a trust region optimization algorithm and I would like to compare it against already done similar work, where authors measures performance on this problem. $$ \min_{u,\gamma}\Bigg\{ ...
2
votes
5answers
34 views

Is it possible to factor a quadratic equation when $a$, $b$, and $c$ are all equal?

I have the equation $4x^2+4x+4$ to factor. I know that need to start with $$(2x \quad )(2x \quad )$$ to make $4^2$, but I can't seem to factor the rest of the way. What should I do?
3
votes
4answers
81 views

Condition for common roots of two Quadratic equations: $px^2+qx+r=0$ and $qx^2+rx+p=0$

The question is: Show that the equation $px^2+qx+r=0$ and $qx^2+rx+p=0$ will have a common root if $p+q+r=0$ or $p=q=r$. How should I approach the problem? Should I assume three roots $\alpha$, ...
1
vote
1answer
30 views

Can anyone answer this with the steps of Assumption > Given > Formula > Solution?

The height in feet of a bottle rocket is given by the function $h(t)=160-16t^2$, where $t$ is the time in seconds. How long will it take for the rocket to return to the ground? What is the height of ...
1
vote
0answers
23 views

Get parameters for given point on quadratic bezier triangle

I have a 2 dimensional quadratic bezier triangle described by the position of its corners $v_0$, $v_1$ and $v_2$ and a handle for each side $h_0$, $h_1$ and $h_2$. The parametric equation with the ...
2
votes
0answers
51 views

How do you the roots of functions that are not quadratics?

I was asked to consider the equation $(x-3)(x+3)^2=c$ I have been asked to find the values of C in which the equation has: three distinct roots only one real root a double root and a single root ...
1
vote
2answers
25 views

Quadratic equation with several variables

How does $$y^{2} - 4y -t^{2} - C = 0$$ Become $$y = 2 \pm \sqrt{t^{2} +2C + 4}$$ I know its the quadratic formula but I dont know how it got it that point The original equation is $$\frac{dy}{dt} ...
2
votes
3answers
85 views

In what conditions a quadratic function has an integer value of $f(x)$ where $x$ is also an integer?

EDITED Sorry, the question was wrong. Please forgive me for this. Suppose a quadratic function $f(x) = ax^2+bx+c$, what I want to know is if in an integer $x$, say $x=1, x=2, x=3, ...$, the function ...
0
votes
1answer
35 views

quadratic equation with imaginary part

How can we solve this equation $x^2-bx-c=0$ where c is not real number. I tried to solve the above equation but I am not sure if it is correct or not. $x_{1,2}=\frac{b}{2}\pm ...
0
votes
1answer
21 views

Two quadratic equations to get an expression

If $a (p+q)^2+2apq+c=0 $ and $a (p+r)^2+2apr+c=0 $ then find $ q.r $ in terms of $ p, a, c$ My try: I tried to equate both equations and got the relations that either $ a=0, q=r, q+r+4p=0 $ Then I ...
1
vote
3answers
57 views

find total integer solutions for $(x-2)(x-10)=3^y$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. How many integer solutions ($x$, $y$) are there of the ...
1
vote
4answers
83 views

Simultaneous equations, $\frac{1}{x}+\frac{1}{y}=1$,$x+y=a$,$\frac{y}{x}=m$

By eliminating $x$ and $y$ from the following equations, I need to find the relation between $m$ and $a$. \begin{align*} \frac{1}{x}+\frac{1}{y}=1 \\ x+y=a \\ \frac{y}{x}=m \end{align*} I tried ...
3
votes
1answer
82 views

reference on $\sqrt{ax}+\sqrt{by}=c$ as a parabola?

Does anyone have a reference on the equation $$\sqrt{ax}\,+\sqrt{by}=c\ ?$$ Clearing square roots and rearranging gives $$ax+by = \frac{(ax-by)^2+c^4}{2c^2}$$ This is the equation of a parabola, so ...
2
votes
2answers
39 views

Trigonometric equation cos sin and power

The problem is $2\cos t - 3\sin^2t +2 = 0$. I get to $2\cos t -3\sin^2t =-2$ I think that I need to use a trigonometric identity like $\cos(x+y)$ and to divide $2\cos t -3\sin^2t$ with the ...
0
votes
2answers
31 views

Nature of Roots of quadratic $af(x) = (x^2+2)(a-1)$

I need help with the following. The problem is stated like so: "The value of the constant $a$ is such that the quadratic function $f(x) \equiv x^2 +4x + a +3$ is never negative. Determine the nature ...
3
votes
2answers
60 views

Solve $\begin{cases} x + y + z = 2 \\ 2xy - z^2 = 4 \\ \end{cases} $ for x, y, z.

It came to my mind to rewrite the expression above as $$\begin{cases} x + y = 2 - z \\ 2xy = (2 - z)^2 + 4z \\ \end{cases} $$ and see if there any restrictions on the values of the variables occur. ...
-1
votes
2answers
29 views

Find equation of a line intersecting parabola at one point only?

Find equation of a line with gradient equal to $2$ which intersects the parabola $y = 6 − x − x^2$ at one point? I tried using the equation of the line $y=2x+c$ and making it equal to $y= -x^2 - x ...
2
votes
1answer
20 views

Finding original price of Tea per kg

A reduction of $ 2 per kg enables a man to purchase 2 kg more tea for $8. Find original price of tea per kg Attempt Let price be $x per kg of tea .So let man buys 10 kg of tea .So total cost is ...