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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2
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3answers
50 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
4answers
66 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 ...
2
votes
1answer
48 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
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)= ...
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
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 ...
0
votes
2answers
26 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 ...
10
votes
2answers
161 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 ...
1
vote
0answers
53 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 ...
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 ...
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 $N(6)$?
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votes
2answers
52 views

The equation $ x^2+x (2x+p)+ 3=0$ has equal roots, find the possible value of p. [closed]

I am having trouble completing this problem
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 - ...
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. $$(x^2-5x+5)^{x^2-36} =1$$
1
vote
0answers
27 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 ...
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: $x+\dfrac{1}{2}=\dfrac{2x+1}{2}$ ...
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 ...
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 ...
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.
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 ...
-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 ...
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 ...
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 ...
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 ...
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 ...
5
votes
5answers
141 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 $\beta<1$ ...
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 ...
-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 ...
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 ...
7
votes
6answers
511 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 ...
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 ...
-1
votes
0answers
18 views

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

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
77 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
78 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
53 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
71 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
80 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", ...