Questions on quadratic equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-h)^2+k$ or $y=a(bx+c)(cx+d)$.

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4
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7answers
93 views

How to find $x^2 - x$?

I'm quite a novice when it comes to maths. I'm on a problem in which I have had to isolate $x$ , through factorials which I completed without problem. However, now I am stuck on a seemingly more minor ...
0
votes
1answer
51 views

Proof of axis of symmetry [duplicate]

How do you prove -b/2a the Axis of symmetry equation using the Quadratic formula?
3
votes
2answers
298 views

Quadratic Formula Question

Concerning the quadratic formula. What does it mean if $b^2-4ac>0$, $b^2-4ac<0$, and $b^2-4ac=0$?
0
votes
1answer
171 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 ...
0
votes
2answers
44 views

What are the parameters of a parabola

In the following figure I understand the $bx+c$ part. It is simply the equation of a line. But I don't understand where did $ax^2$ came from? What exactly is it? What does $a$ tell us about a ...
2
votes
2answers
51 views

Find the value of $a$ such that at least one root of the equation $f(x)=x^2 - (a-3)x + a =0$ is greater than $2$.

As the title says. To solve this problem I took two cases and solved them separately: 1. when the $x$ coordinate of the vertex is greater than $2$ and $f(2)>0$; 2. when $f(2)<0$. However I ...
1
vote
2answers
80 views

Find maximum of a system of equations

You have 300 meters of fencing with which to build two enclosures. One will be a square, and the other will be a rectangle where the length of the base is exactly twice the length of the height. (a) ...
5
votes
4answers
164 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
3
votes
2answers
3k views

Show quadratic equation has two distinct real roots.

$x^2 - (5-k) x + (k+2) = 0 $ has two distinct real roots. So, in the markscheme of this question, they take the discriminant ($-b^2 + 4ac$) and say it is greater than 0. That is, $( (-(5-k)^2 - ...
0
votes
5answers
79 views

How do you factor a quadratic expression, without using the formula?

I am asked to factor $2x^2 -3x+1=0 $ using factorization, but I run into fractions, and it becomes very messy and complicated to deal with, especially since specifically asked not to use the formula. ...
1
vote
4answers
162 views

How can I solve equation $x^2 - y^2 -2xy - x + y = 0$?

I have this equation with 2 variables - $$x^2 - y^2 -2xy - x + y = 0$$ The only condition I have is that $x + y$ should be greater than $10^{12}$. EDIT - I need $x$ and $y$ to be integer. I ...
3
votes
3answers
324 views

Algebraic Relationships - Quadratic Equations

I am having a tough time with the following question: If $x$ is real and $p=3(x^2 + 1)/(2x-1)$, then prove that $p^2 - 3(p+1)\geq 0$. I don't know how to tackle this question. Thanks for your ...
0
votes
0answers
54 views

System of quadratic diophantine equations 2

I am looking for a way to simultaneously transform the following four expressions into perfect squares, $1+x_1^2, 1+x_2^2, 1+x_3^2, x_1^2+x_2^2+x_3^2$, i.e. I want to find a rational parametrization ...
0
votes
4answers
112 views

Completing the square with second degree coefficient greater than one

How do I complete the square when the second degree coefficient is greater than one. I can do it when $x^2+4x-4=0$, for example, but I can't work out how to do when $3x^2+4x-4=0$.
6
votes
3answers
128 views

How to solve the following? $ x^3+1=2{(2x-1)}^{1/3} $.

Find all the real solutions of $$x^3+1=2{(2x-1)}^{1/3} $$ I tried to cube both sides but got messed up with a nine degree equation! Please help. Thanks in advance!
0
votes
4answers
62 views

Quadratic equations and probability

The inequality: 4p^2-17p+4>0 Solving using quadratic equation: (−(−17)±√(−17)^2−4⋅4⋅4)/8 =(12±√225)/8 I realize why p = 4 or p = 1/4, and in this case p represents and probability so the solution ...
0
votes
1answer
43 views

Finding the value of $n$ in this question?

Find the value of n for which the quadratic equation $$ \sum_{k=1}^{n}(x+k-1)(x+k) =10n $$ has solutions $α$ and $α+1$ for some $α$.
0
votes
1answer
54 views

Inequality involving a quadratic equation

Let $a,b,c$ be integers and suppose the equation $$f(x) = ax^2 + bx + c = 0$$ has an irrational root $r$ . Let $u=\dfrac{p}{q}$ be any rational no. such that $|u-r|<1$. Prove that ...
-1
votes
2answers
63 views

Quadratic equations question

let $P(x)$ and $Q(x)$ be 2 quadratic eqs. with one non-rational root common and integral coefficients. prove $P(x) = r.Q(x)$, for some rational no. $r$ TRIED ANSWER: Let $P(x)= ax^2 + bx + c$ ...
3
votes
2answers
83 views

If $P(x) = ax^2 + bx + c$ and $Q(x) = -ax^2 + dx + c$, then prove that $P(x) \cdot Q(x) = 0$ has at least two real roots?

How should i solve the same? I assumed the roots be $ \alpha, \beta $ for $ P(x) $ and $ \gamma, \delta $ for $ Q(x) $. Product of roots turn out to be of the opposite signs, being $$ \alpha \cdot ...
1
vote
1answer
28 views

Find values of the parameter a so that equation has equal roots.

$x^2+2a\sqrt{a^2-3}x+4=0$ My final result was 2 and -0.5. Was it correct?
0
votes
2answers
71 views

Prove $\alpha^3 + \beta^3 = S^3 - 3PS$ in Quadratic Equation

We have a quadratic equation like this: $ax^2 + bx + c = 0$ and we know that $S = \alpha + \beta = -b/a$ and $P = \alpha\beta = c/a$. How we can prove that $\alpha^3 + \beta^3 = S^3 - 3PS$ and is ...
0
votes
1answer
52 views

Quadratic equation problem. Composition of functions

Suppose $p(x)$ and $q(x)$ are quadratic polynomials and the three largest roots of $p(q(x))$ are $10$, $20$ and $23$. What is the smallest root of $p(q(x))$? Then, there will be 4 roots. $q(10)$ ...
0
votes
1answer
41 views

Request for help with a quadratic polynomial question.

If the rooots of the equation $x^2+bx+c=0$ are real , show that the roots of the equation $x^2 +bx+c(x+a)(2x+b)$ are again real for every real number a. I assumed the discriminant of the first ...
2
votes
2answers
50 views

Fitting a quadratic polynomial to two points such that it is always concave downward

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, I'd like to construct a quadratic polynomial of the form $y = a_2x^2 + a_1x + a_0$ such that it intersects both points and is concave downward (i.e., ...
1
vote
3answers
28 views

How to invert these equations

Apologies in advance as maths has never been my strong point (I'm not even sure which tag to use). I'm developing some software that uses some equations to convert values being read from a hardware ...
0
votes
1answer
78 views

Isolate “a” in a quadratic function

You have a quadratic function: $ax^2 + bx + c = y$. If you know $b$ and $c$, are able to plug any domain value $x$ into this blackbox equation and receive a range value $y$, and do not know the vertex ...
1
vote
1answer
130 views

quadratic function of inanimate sphere

A life form standing on the surface of an unknown planet throws a small inanimate sphere vertically upwards and then steps backwards. The life form releases the sphere at a height of 1m; after 4 ...
0
votes
1answer
56 views

Is there any solution to this quadratic Diophantine equation?

Can one find all positive integer triplets $(x,y,z)$ satisfying this parametric equation : $$ax^2 + (1-a)x + by^2 + (1-b)y = cz^2 + (1-c)z$$ Here, $a, b, c$ are positive integers greater than $1$. ...
0
votes
0answers
17 views

Quadratic function as permutation of sequence

Say I have a $n \in \mathbb{N}$ and $$a_i := (1,2,...,2^n)$$ and two function $$f(i) = \sum_1^i i = \frac{i(i+1)}{2}$$ $$g(i) = f(i) mod 2^n$$ When I now look at a new sequence $$b_i = (a_{g(0)}, ...
2
votes
1answer
43 views

Find the real parameter a so that the equation has real and positive roots

My problem is: $(2-x)(x+1)=a$ How do I find the value of the parameter $a$ so that the equation has real and positive roots?
0
votes
1answer
507 views

How to find quadratic function in vertex form from two points?

I'm starting to learn about quadratic formulas in math class. This question came up in a homework packet: A WNBA player takes a three-point shot 22 feet away from the basket, The ball reaches ...
1
vote
3answers
32 views

Finding roots of complex quadratic equation

I'm trying to solve for the following equation: $$|(1+50*i*x)^2|$$ I keep getting the form $$-2500x^2 + 100ix + 1 $$ when the problem needs to have the following form: $$2500x^2 + 1$$ What steps ...
2
votes
2answers
67 views

Solving a fractional quadratic equation problem by completing the square

I have the following problem to solve using the method of completing the square. $$2x^2-3x-1 = 0$$ Here is where I've gotten to so far on this problem. $$2x^2-3x = 1$$ $$x^2-\frac{3}{2}x = ...
1
vote
1answer
44 views

how should i go about solving the following problem??

$f(n)=a^n-b^n$ where $a$ and $b$ are roots of the following equation .$$5x^2-2x+1=0$$ Then find the value of $$\frac{5f(10)+f(9)}{f(8)}$$ I realised we can use the 5 in the equation as $\frac{1}{ab}$ ...
0
votes
2answers
36 views

Prove that $g(x) > 0$

If $f(x)$ is a quadratic expression such that $f(x) > 0,\ x\in\mathbb{R}$ and if $g(x)= f(x) + f'(x) + f''(x)$, then prove that $g(x) > 0, \ x\in\mathbb{R}$.
4
votes
4answers
118 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
0
votes
1answer
12 views

Quadratic expression

If $$\frac{a_0}{n+1}+\frac{a_1}{n}+\frac{a_2}{n-1}+\ldots+\frac{a_{n-1}}{2}+a_{n}=0,$$ then the maximum possible number of roots of the equation ...
0
votes
3answers
33 views

Why the discriminant determine whether a quadratic has real roots or not?

It's been quiet a mystery for, why is this true:? If $\Delta>0$ then it have two solutions. If $\Delta=0$ then it have only one solution. If $\Delta<0$ then it have no solutions
7
votes
1answer
71 views

Prove that $n$ is divisible by $6$

Problem: Let $x^2+mx+n$ and $x^2+mx-n$ give integer roots where $(m,n)$ are integers. Show that $n$ is divisible by $6$ My attempt: Since the roots are integers then the discriminants of both the ...
1
vote
2answers
66 views

Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
1
vote
2answers
35 views

Analytical approach to a quadratics problem

I'm a bit rusty on functions and this exercise got me thinking quite a bit. The function $y=x$ is tangent to the graph of a certain $g$ function in $x=0$. Function $g$ can be defined as: ...
1
vote
2answers
772 views

How to solve a quadratic equation with two unknowns?

I know how to solve quadratic equations when there's only one unknown, but I'm a bit confused on what I do if I have $2$ unknowns e.g: $$x^2 + 2kx + 81 =0$$ With just $x^2 + 2kx + c=0$, obviously ...
0
votes
1answer
22 views

$2$nd degree inequality question

If I have an inequality of the second degree, can I solve it using the quadratic formula? Example: $$-t^2+48t+100>500$$ Can i solve it by doing: $$-t^2+48t+(100-500)=0$$ and apply the quadratic ...
3
votes
2answers
66 views

Evaluate $a+b+c+d$

If $a$, $b$, $c$, and $d$ are distinct integers such that $$(x-a)(x-b)(x-c)(x-d)=4$$ has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$? I tried to analyze graphically by ...
2
votes
2answers
71 views

Find the value of $\left | b-c \right |$

Given that $a, b, c \in \mathbb{Z}$, $a>10$ and $$(x-a)(x-12)+2=(x+b)(x+c)$$ Find the value of $\left | b-c \right |$ NOTE: The answer to this problem (as given on the last page of my book) is ...
0
votes
0answers
30 views

Convergence rate of $x_{k+1}=3x_k^2/n+3$

I've found the following claim in a slightly different form here (page 4, bottom of the left column) Starting from $x_0\le n/3$, the recurrence equation $$3\le ...
1
vote
2answers
64 views

Prove that $|f(x)| \le \frac{3}{2}$ when $f(x)=ax^2+bx+c$ [duplicate]

Suppose $f(x) = ax^2+bx+c$ where $x \in [-1,1]$. If $f(-1),f(0),f(1)\in [-1,1]$ show that $|f(x)| \le \frac{3}{2}$ $\forall x \in [-1,1]$. This is how I tried: $f(0)=c$ $f(1)=a+b+c$ $f(-1)=a-b+c$ ...
0
votes
2answers
15 views

Let f be a continuous function defined on [-2009,2009] such that f(x) is irrational for each $x \in [-2009,2009]$ …

Problem : Let f be a continuous function defined on [-2009,2009] such that f(x) is irrational for each $x \in [-2009,2009]$ and $f(0) =2+\sqrt{3}+\sqrt{5}$ Prove that the equation $f(2009)x^2 +2f(0)x ...
0
votes
1answer
63 views

Confusion regarding Quadratic equations and RHS = 0

Recently, I am becoming confused with how it is said that in a quadratic equation you MUST make the RHS $ = 0$. But I am stumbling across many equations where it is calculated (The following is for ...