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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6answers
156 views

Find the roots of quadratic polynomial given one root of another quadratic polynomial?

if $a,b,c$ are Real numbers and $1$ is a root of equation $ax^2+bx+c=0$ then curve $y = 4ax^2+3bx+2c$ , (a is not zero) intersects $x$ axis at how many points? I get a relation $a+b+c = 0$ I tried ...
0
votes
1answer
34 views

Determining the number of roots

Given a set of two equations (one linear and one quadratic in $x$ and $y$) as follows:- $$ax + by + c = 0 \tag 1$$ $$Ax^2 + Bxy +Cy^2 + Dx + Ey + F = 0 \tag 2$$ What are the conditions that can be ...
1
vote
0answers
45 views

Taylor's formula and its quadratic term

I struggle with the following problem: For a function $$f: \mathbb{C} \rightarrow \mathbb{R}~,$$ $f$ attains its maximum for $z_0= e^{i\pi/3}$, $f(z_0)=F_{max}.$ Assume we may use Taylor's theorem ...
0
votes
1answer
25 views

Linear-Equation - What is the error in this equation?

I want to know if this equation is wrong? \begin{align} 4(x - 3) + 2(x - 1) - 3 &< 10\\ 4x-12 + 2x-1-3 &< 10\\ 4x - 2x &< 10 + 12 + 1 + 3\\ 2x &< 26\\ ...
0
votes
2answers
204 views

How to use the property $\sqrt{a}\sqrt{b}=\sqrt{ab}$ with caring that it not always holds?

I was solving a question from my book. If $\alpha$, $\beta$ are the roots of $pt^2+qt+q=0$, $p \neq 0$ and $q \neq 0$ then show that, ...
1
vote
1answer
22 views

Troubleshooting Textbook: Using Generating Functions for Non-Homogenous Recurrence

I am learning about Generating Functions to solve non-homogenous linear recurrences, and I can't seem to get the right solution, no matter what I do. Also, the example I have to work off of in the ...
0
votes
1answer
27 views

Forming Quadratic equation from roots

I think I just need some background. I've got the following quadratic equation: $$ 1 - x - 2x^2 = (1-2x)(1 + x) $$ But if I solve it with the quadratic equation, I get the roots: $$ \frac{1 + ...
1
vote
2answers
31 views

Simplifying a solution to a quadratic equation

I am solving: $(\sigma_A^2 - 2\rho\sigma_A\sigma_B +\sigma_B^2)x^2 +2(\rho\sigma_A\sigma_B - \sigma_B^2)x +\sigma_B^2 = 0$ I need to show that a real $x$ exists if and only if $\rho = \pm 1$ Using ...
1
vote
5answers
30 views

Existence of rational roots in a quadratic equation

Consider the quadratic equation $(a+c-b)x^2 + 2cx + b+c-a = 0 $ , where a,b,c are distinct real numbers and a+c-b is not equal to 0. Suppose that both the roots of the equation are rational . Then a) ...
0
votes
1answer
92 views

Existence of a common root among two quadratic equation

The equations $x^2 + x + a = 0$ and $x^2 + ax+ 1 = 0$ a) Cannot have a common real root for any value of a b) have common real root for exactly one value of a c) have a common real root for exactly ...
1
vote
1answer
54 views

What are the solutions for $2^x=x^2$? [duplicate]

What are the solutions for $2^x=x^2$? I noticed there were 2 roots: $2,4$. Are there any other roots, and how do you calculate them?
3
votes
1answer
75 views

Trying to prove $c^3a^2+(9c^2-b^2)a+(27c-10b)=0$ has no positive integer solutions

I'm trying to prove (or, I suppose, disprove) the following claim, in either version. Conjecture (Strong Version): There are no positive integers $a,b,c$ such that $$c^3a^2+(9c^2-b^2)a+(27c-10b)=0.$$ ...
2
votes
2answers
42 views

If the range of the function $f(x)=\frac{x^2+ax+b}{x^2+2x+3}$ is $[-5,4],a,b\in N$,then find the value of $a^2+b^2.$

If the range of the function $f(x)=\frac{x^2+ax+b}{x^2+2x+3}$ is $[-5,4],a,b\in N$,then find the value of $a^2+b^2.$ Let $y=\frac{x^2+ax+b}{x^2+2x+3}$ $$x^2y+2xy+3y=x^2+ax+b$$ ...
1
vote
2answers
45 views

Let $f(x)=\sqrt{\frac{x^2+ax+4}{x^2+bx+16}}$ is defined for all real $x$,then find the number of possible ordered pairs $(a,b),$

Let $f(x)=\sqrt{\frac{x^2+ax+4}{x^2+bx+16}}$ is defined for all real $x$,then find the number of possible ordered pairs $(a,b),$ where $a,b$ are both integers. As $f(x)$ is defined for all real ...
3
votes
4answers
78 views

Can every parabola be written in the form of a quadratic $y=ax^2+bx+c$ or $x=dy^2+ey+f$?

I understand that the graph of any equation of the form $y=ax^2+bx+c$ is a parabola (please correct me if I am mistaken). My question is about the converse: Can every parabola be written in the form ...
1
vote
2answers
38 views

How to find the unknown in this log inequality??

Find all values of the parameter a $\in\Bbb R$ for which the following inequality is valid for all x $\in\Bbb R$. $$ 1+\log_5(x^2+1)\ge \log_5(ax^2+4x+a) $$ I'm lost when I got to this stage: $ ...
0
votes
1answer
42 views

If the roots of $ax^2+bx+c=0$ are of the form $\frac{m}{m-1}$ and $\frac{m+1}{m}$ then find..

Problem : If the roots of $ax^2+bx+c=0$ are of the form $\frac{m}{m-1}$ and $\frac{m+1}{m}$ then find the value of $(a+b+c)^2$ My approach : Let $\alpha, \beta$ are the two roots of the given ...
1
vote
2answers
30 views

Expansion and factorisation

I have a little problems with a few questions here and I need help.. Thanks ... Factorise completely $$9x^4 - 4x^2 - 9x^2y^2 + 4y^2 $$ My workings .. $$ (3x^2+2x)(3x^2-2x) - y^2 (9x^2-4) = ...
0
votes
1answer
21 views

From the graph of a quadratic equation, find the range of values of $x$.

The equation of the graph is $$ y = -x^2 + 9x - 18 $$ From the graph sketched, find the range of values of $x$ for which $x^2 + 18 > 9x$. Workings $$ y = -(x-3)(x-6) $$ I'm not sure what is ...
1
vote
3answers
51 views

If $ax^2+bx+c \leq p(x) \leq lx^2+mx+n$ , show that the degree of $p(x)$ is $2$.

If $a,l\neq0$ , $ax^2+bx+c \leq p(x) \leq lx^2+mx+n$ , show that the degree of $p(x)$ is $2$. How can we exactly say (how to prove) that $p(x)$ is a quadratic ? What methods can be used to ...
0
votes
1answer
32 views

Does any quadratic function in the form $an^2 + bn + c$ equal $\Theta(n^2)$ in asymptotic notation?

On a Khan Academy post (see here) about Big-$\Theta$ notation, the author attempted to convert the quadratic function $6n^2 + 100n + 300$ to asymptotic notation. They started by dropping the $n^2$ ...
0
votes
1answer
36 views

solving an equation $x^x= c$ [duplicate]

I would like to find a solution $x$ for $x^x = c$ where $c$ is a positive constant. Firstly I'm looking for an approximative solution when $c$ tends to infinity. Thank you in advance
0
votes
2answers
23 views

Quadratic equations - Fastest way to find the value of d

$\frac{2d^2-d-10}{d^2+7d+10} = \frac{d^2-4d+3}{d^2+2d-15}$ What is the optimal solution for finding the value of d?
0
votes
4answers
61 views

Find the real values of $P$ for which $f(x)=P$ has exactly one solution.

$$f(x)=(x-1)^2(x-2)+1$$ Find the real values of $P$ for which $f(x)=P$ has exactly one solution. Hi, I'm a little bit confused with this question.I don't know how I should start. I think I ...
9
votes
3answers
527 views

Quadratic Formula, nature of roots with Trigonometric Functions

The original problem: If $0\le a,b\le 3$ and the equation $$x^2+4+3\cos(ax+b)=2x$$ has at least one real solution, then find the value of $a+b$ $$$$ At first, on rearranging, I got the ...
1
vote
0answers
38 views

If you are using piecewise quadratic polynomials to approximate the function $f (x) = \ln x$ on the interval $[1, 2]$

If you are using piecewise quadratic polynomials to approximate the function $f (x) = \ln x$ on the interval $[1, 2]$ and expect the maximum error to be smaller than $10^{-6}$, how many subintervals ...
1
vote
3answers
41 views

Calculating range of values of $k$ s.t. the graph $y=4x^2-kx+25$ does not cut or touch the $x$ axis

Calculate the range of values of $k$ so that the graph $y=4x^2-kx+25$ does not cut or touch the $x$ axis. I just don't know what to set delta to as I can't work out if the graph would be a ...
1
vote
1answer
50 views

Quadratic Equation; Roots' Magnitude Less than 1

What are the conditions on $a$ and $b$ so that the roots (real or complex) of the equation have magnitude $< 1$. $$λ^2 − (a − b + 1)λ + a = 0$$ On a separate note, if you could explain (NOT ...
4
votes
2answers
45 views

Find the value of $\sqrt[4]{\alpha}-\sqrt[4]{\beta}$,where $\sqrt[4]{.}$ denotes the principal value.

If $\alpha$ and $\beta$ are the roots of the equation $x^2-34x+1=0$,find the value of $\sqrt[4]{\alpha}-\sqrt[4]{\beta}$,where $\sqrt[4]{.}$ denotes the principal value. I found out the $\alpha$ ...
1
vote
5answers
38 views

quadratic equation expressed as a range of values

Solve $6 - x - x^2 < 0$ A. $-3 < x < 2$ B. $x < -3, x > 2$ C. $-2 < x < 3$ D. $x < -2, x >3$ So this is a very easy quadratic to solve: $x^2 + x - ...
1
vote
1answer
63 views

Resolving the tedious cubic

The equation given to me is $$4x^4 + 16x^3 - 17x^2 - 102x -45 = 0$$ I'm asked to find it's resolvent cubic which is not so difficult to find. But the problem is that the question further asks to find ...
4
votes
2answers
50 views

Maximize the sum $a+b$

If the equation $x^4-4x^3+4x^2+ax+b=0$(where $a,b$ are reals) has $2$ distinct positive roots $\alpha, \beta$ such that $\alpha+\beta =2\alpha \beta$, then find the maximum value of $a+b$. I have ...
3
votes
2answers
36 views

If $a\in R$ and the equation $-3(x-\lfloor x \rfloor)^2+2(x-\lfloor x \rfloor)+a^2=0$ has no integral solution,then all possible values of $a$

If $a\in R$ and the equation $-3(x-\lfloor x \rfloor)^2+2(x-\lfloor x \rfloor)+a^2=0$ has no integral solution,then all possible values of $a$ lie in the interval $(A)(-1,0)\cup(0,1)$ $(B)(1,2)$ ...
-1
votes
2answers
23 views

Express $s = -5t^2 + 40t$ in the form of $a(t-b)^2 + c$, where $a$, $b$ and $c$ are the constants.

$s= -5t^2+ 40t$. Express $s$ in the form of $a(t-b)^2 + c$, where $a$, $b$ and $c$ are the constants. $s = -5t(t-8)$. I have factorized it.
0
votes
5answers
62 views

Solving $ \sqrt{5x+1}+\sqrt{x-1}=2$

How to solve: $$ \sqrt{5x+1}+\sqrt{x-1}=2$$ I can tell that 1 is a solution but I am not sure how to solve this algebraically, do i start by squaring both sides?
4
votes
2answers
117 views

How can we prove that a quadratic equation has at most 2 roots?

A quad equation can be factored into two factors containing $x $, but how can we prove that there no other sets of different factors yielding OTHER VALUES OF $X $?
1
vote
1answer
104 views

Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x - 9)^2 + (y - 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that ...
1
vote
1answer
48 views

Given a set of $(x,y)$ coordinate pairs how can I come up with the equation for a curve?

Say for example I have a set of coordinate pairs as follows: x y 0 100 20 90 100 0 I would like to generate an equation for this curve In the ...
0
votes
2answers
45 views

The quadratic spline

I'd like to fit the data in table as blow x f(x) 3.0 2.5 4.5 1.0 7.0 2.5 9.0 0.5 when $x=5$, I want to find value of $f(x)$ by using ...
2
votes
1answer
42 views

Least Squares Fitting Quadratic Equation to a set of points

My math skills from my college days are a bit rusty, so if my terminology is wrong, I apologize. I will try to be as clear as I can. I have a set of 50 points on a plane that roughly follow the ...
0
votes
1answer
13 views

How to solve for a & b in a parabolic equation?

Not a mathematician at all here, actually a programmer and this came up... I need some help on something that is pretty basic I'm guessing. Given the following formula how would I go about solving ...
0
votes
5answers
84 views

Factorizing quadratics mentally

How would it be possible to factorize quadratics mentally, for example the following one? $$2x^2+7x+3$$ Maybe even something like $$3x^2+22x+24$$
0
votes
1answer
14 views

Finding parameter for quadratic equation

Given $x^2 - 3ax + a^2 = 0$ and $$\frac{x_1^4-x_2^4}{\sqrt{5}x_1x_2} + x_1 + x_2 -20x_1x_2 - 4 = 0$$ Find $a$. The answer is $1$ ($a = 1$) I tried to present $x_1^4 - x_2^4$ as ...
2
votes
3answers
45 views

Why is the graph of a quadratic function a parabola?

I'm sorry for the stupid question, but it seems that extensive googling didn't yield an answer. I've learned about parabolas, and how the parabola is the curve that is equidistant from a point ...
1
vote
0answers
67 views

If $a+b+c = 0$ then the quadratic equation $3ax^{2}+2bx +c=0$ has atleast one root in _________?

If $a+b+c = 0$ then the quadratic equation $3ax^{2}+2bx +c=0$ has atleast one root in _________? Rolle's theorem states that if $f(a) = f(b)$ then there exists a $p \in [a,b]$ such that : $f'(p) = ...
6
votes
2answers
61 views

Quadratic equation with one root in $[0,1]$ and other root in $[1,\infty]$

Find the values of $a$ for which $x^2-ax+2=0$ has one root in $[0,1]$ and other root in $[1,\infty]$. The twoo rots are $$\frac{a\pm\sqrt{a^2-8}}{2}$$ The smaller root should be less than $1$. So ...
1
vote
4answers
79 views

How would one solve the following equation?

This equation is giving me a hard time. $$e^x(x^2+2x+1)=2$$ Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph ...
2
votes
2answers
40 views

Quadratic expression problem involving sides of a triangle

If $a,b,c$ be the sides of a triangle where $a\neq b\neq c$ and $\lambda \in R$, if roots of the equation $x^{2} + 2\lgroup a+b+c\rgroup x + 3\lambda\lgroup ab+bc+ca\rgroup =0$ are real, then what is ...
1
vote
2answers
50 views

Pairs of Quadratic equations

If each pair of the three equations $$x^2 + P_1x + q_1 = 0$$ $$x^2 + P_2x + q_2 = 0$$ $$x^2 + P_3x + q_3 = 0$$ have a common root, then prove that $$P_1^2 + P_2^2 + P_3^2 + 4(q_1 + q_2 + q_3) = 2 ( ...
0
votes
1answer
43 views

How to graph a Quadratic equation.

I have an equation that goes: $0.0001x^2 - 0.22x + 197$. I'm not asking for the answer, but instead, how can I graph it without dealing with these insanely tough numbers.