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 $y=a(x+b)(x+c)$.

learn more… | top users | synonyms (1)

-2
votes
1answer
29 views

Solve the quadratic equation $(2-y)^4=3(2-y)^2+1$

Solve $$(2-y)^4=3(2-y)^2+1$$ The answer is supposed to be $y=4\pm \sqrt{6+\frac{13}2}$. I have tried to work this problem out but I cannot get the answer that is in the book.
-1
votes
1answer
20 views

Standard Form of a Parabola [on hold]

Given the vertex, V (2,0) and the intercepts (3,2); How can I get the standard form using these given? Please show the solutions. My top concern is that, how can I get, a. When the standard form is ...
1
vote
1answer
30 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 ...
8
votes
4answers
645 views

Simple Trig Equations - Why is it Wrong to Cancel Trig Terms?

In the following problem, I first did it using a cancellation of $sin^2\theta$, working shown below, which gave the wrong answer. Having looked at the question again, I saw it could be solved by ...
0
votes
3answers
27 views

Finding remainder

Okay I saw this one on a test so here it goes: A polynomial of (degree > 3) when divided by $ (x-1)^2$ and $x-3$ leaves the remainder $2x+1$ and $15$ respectively. The remainder when it is divided by ...
10
votes
2answers
181 views
+50

Find all pair of cubic equations

Find all pair of cubic equations $x^3+ax^2+bx+c=0$ and $x^3+bx^2+ax+c=0$, where $a,b$ are positive integers and $c$ not equal to $0$ is an integer, such that both the equations have three integer ...
2
votes
1answer
55 views

How to solve these equations?

How to solve these equations for a, b, c and x? I have the following: $ 2a+b+c = 1$ $a = (a+b)x + 0.25(a+c) $ $a=(a+c)(1-x)$ $b=a(1-x)+c(x-0.25)$ $c=b(1-x)+a(x-0.25)$ I tried, but ended ...
0
votes
4answers
68 views

Highschool Algebra: $n^2 = 18n$?

I'm beginning to get into maths outside of school and at the moment I'm refreshing myself on the basics which explains why this question appears to be so simple. I formulated this equation to find ...
1
vote
0answers
55 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations ...
1
vote
0answers
22 views

Find first positive perfect square in polynomial time

I have a quadratic. for example $$1x^2+6884x+3297$$ Is it possible to find the first perfect square in the series in polynomial time where both x and y are whole positive integers. In the above ...
0
votes
2answers
22 views

Need help with tangents to a quadratic

The quadratic $y=kx^2+(3k-1)x-1$ and the straight line $y=(k+1)x-11$ meet. Find the range of value(s) of $k$ such that the line is a tangent to the curve. Got this question for school. Seems really ...
5
votes
3answers
102 views

Range of a Rational Function

How to find the Range of function $$f(x)= \frac{x^2-3x-4}{x^2 - 3x +4}$$ I tried to equate the expression to $y$, then cross multiplied $$ y= \frac{x^2-3x-4}{x^2 - 3x +4}$$ $$ y(x^2 - 3x +4)= ...
0
votes
2answers
71 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 ...
0
votes
1answer
33 views

Solving an equation containing 4th power of variable.

I know how to solve Quadratic equations. Recently i came across the equation of type $ax^4 + bx^2 + c = 0$ and i had to solve it. So what i did is that i supposed $x^2 = y$ so that the above equation ...
4
votes
2answers
49 views

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ ...
0
votes
2answers
28 views

Find the values of a, b and c

The question: Find values of $a,b,c.$ if $\displaystyle \frac{x^2+1}{x^2+3x+2} = \frac{a}{x+2}+\frac{bx+c}{x+1}$ My working so far: http://i.imgur.com/VegifVa.jpg How do I isolate $a$, $b$ and ...
5
votes
5answers
142 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 $\beta<1$ ...
15
votes
3answers
2k views

How to solve equations to the fourth power?

Is it possible to manually retrieve the value of $y$ from the following equation $$153y^2-y^4=1296$$ WolframAlpha has four solutions for $y$: $-12, -3, 3, 12$. How has it solved? What I've achieved ...
0
votes
1answer
35 views

How to create quadratic equation given $y$ intercept, and maximum and $B=8$?

The given are Two x-intercepts y-intercept(0,-4) Maximum at (2,4) i tried everything i know...its been a long time since I have been doing math problems but the only way i thought about was to use ...
0
votes
1answer
291 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
1
vote
1answer
51 views

Proof related to quadratic equation

Suppose that m and n are integers such that both the quadratic equations $x^2 + mx − n = 0$ and $x^2 − mx + n = 0$ have integer roots. How to prove that n is divisible by 6?
4
votes
2answers
34 views

How can we assume that the all parabolas in the form 'ax^2 + bx +c' are symmetrical?

So I was reading an answer to a question pertaining to the derivation of the line of symmetry. It reads as follows: The vertex occurs on the vertical line of symmetry, which is not affected by ...
1
vote
2answers
33 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 $N(6)$?
0
votes
1answer
30 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 ...
-1
votes
1answer
35 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
vote
2answers
185 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 - ...
4
votes
0answers
115 views

Quadratic equation in $\mathbb{Z}/n\mathbb{Z}$?

I would like to ask for some help about the following problem. Given is a polynomial $\,f(x)=ax^{2}+bx+c\,$ in $\,\mathbb{Z}/ n\mathbb{Z},\,$ we know that this quadratic equation $\,f(x)=0\,$ has ...
1
vote
3answers
69 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. $$(x^2-5x+5)^{x^2-36} =1$$
3
votes
2answers
381 views

$y =f(x) =(ax^2 + bx +c)/(dx^2+ex+f)$ We have to find the conditions for this it takes all real values.

$$ y=f(x)=\frac{ax^2+bx+c}{dx^2+ex+f} $$ We have to find the conditions for this it takes all real values. MY solution One approach is to equate it to y and for a quadratic of x and put discriminant ...
1
vote
4answers
63 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 ...
1
vote
3answers
113 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: $x+\dfrac{1}{2}=\dfrac{2x+1}{2}$ ...
3
votes
1answer
165 views

Proving an equality involving binomial coefficients and summations

Question: $$\sum_{k=0}^{n}\left ( -1 \right )^{k}\binom{2n}{k}\binom{2n-k}{2n-2k}=\sum_{2n}^{k=0}\binom{2n}{k}^{2}\left ( \frac{1+\sqrt{5}}{2} \right )^{2n-k}\left ( \frac{1-\sqrt{5}}{2} \right ...
0
votes
5answers
65 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 ...
2
votes
2answers
61 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.
-2
votes
1answer
30 views

Determine values of $a$ given $c$ in quadratic equation and a point $(1,2)$ [closed]

The point $(1,2)$ is on the graph of the quadratic function $f(x) = ax^2 + bx + 1$. Determine the values of $a$, such that the graph of $f(x)$ intersects the $x$-axis at two distinct points. This ...
1
vote
1answer
48 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 ...
0
votes
3answers
48 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 ...
1
vote
2answers
45 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 ...
1
vote
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 ...
0
votes
1answer
16 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 ...
-1
votes
1answer
13 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 ...
1
vote
3answers
30 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 ...
2
votes
2answers
46 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 ...
1
vote
0answers
384 views

Solving a system of quadratic equations

I'm facing a rather trivial problem which I seem unable to solve... Not being a mathematician (but an engineer with a bit of knack for math), I managed to formulate it in a way that seemed solvable to ...
5
votes
5answers
959 views

find the least a, for which two equations have a common root

Could you help me out please. I have two equations: $2x^2-3x+1=0 $ and $ 2x^2-(a+3)x+3a=0$ I need to find the least $a$ for which these two equations have a common root. At a first glance I thought ...
7
votes
6answers
512 views

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

If $3x^2 -2x+7=0$ then $(x-\frac{1}{3})^2 =$ ? I'm so confused. It's a self taught algebra book. The answer is $-\frac{20}{9}$ but I don't know how it was derived. Please explain. Thanks for ...
0
votes
1answer
29 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 ...
2
votes
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
votes
5answers
54 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 ...