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

learn more… | top users | synonyms

0
votes
1answer
74 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 ...
0
votes
0answers
15 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)}, ...
0
votes
1answer
201 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 ...
2
votes
2answers
62 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
35 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 $g(x) > 0, \ x\in\mathbb{R}$.
5
votes
4answers
110 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
31 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
1
vote
2answers
39 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
34 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: ...
0
votes
1answer
21 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 ...
4
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 ...
3
votes
2answers
62 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
28 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 ...
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
42 views

Quadratics and roots

The question I am trying to solve is this: $4 x^2 - 3 x - 3 = 0$ has roots $p, q$. Find all quadratic equations with roots $p^3$ and $q^3$. I was able to answer this question by simply finding the ...
0
votes
3answers
39 views

Find the set of real numbers ($x$ not equal to zero) such that $2x + 1/x < 3$.

Pretty straightforward question, I just had a question for the conclusion. I rearranged, and factored and have the quadratic: $$2x + 1/x < 3$$ (multiply both sides by x and rearrange) $$2x^x - ...
1
vote
2answers
58 views

Proving a simple equation with complex numbers

Fix $A \in ℂ$ and $B \in ℝ$ Let $z \in ℂ$. Show that the equation $|z^2| + Re(Az) + B = 0$ has solutions iff $|A^2| ≥ 4B$ I have no trouble proving the forward implication, but its the "only if" ...
0
votes
0answers
15 views

How to minimize this quadratic function?

As described at page 3 of this document, I need to minimize the following quadratic function: $E(w,x,y,z) = \sum_i \frac{(w-T_i(x,y,z))^2}{1+|\Delta f(x_i,y_i,z_i)|^2} $ where $w=f(x,y,z)$ and ...
5
votes
2answers
163 views

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$. I made quite a few attempts but could not ...
1
vote
1answer
220 views

Irreducible quadratic factors; partial fraction decomposition.

Please help me understand why there is Dx+E, Fx+G etc, instead of the regular A's, B's, C's etc. What is it about the irreducible quadratic in the denominator that makes it different on top?
1
vote
2answers
61 views

Solve $f(x) = ax^2 + bx + c$ to find the value of $K$

$f(x)=ax^2+bx+c$, where $a=-9$, $b=12$ and $c=16$. If $$-1<f'(x)<1$$ then $h<x<k$. To $2$ decimal places, what is the value of $k$? Hi, this is working for solving $f(x) = ax^2 + bx + ...
0
votes
0answers
14 views

Quadratics and function question

A quadratic function is given by ${h(x) = ax^2 + bx + c}$ where ${a}$, ${b}$, and ${c}$ are all nonzero real numbers. The function ${h(x)}$ intersects the x-axis at two distinct points and satsifies ...
0
votes
1answer
25 views

Quadratic roots question

If $3.5 - {\sqrt 2}$ and $3.5 + {\sqrt 2}$ are the roots of a quadratic equation ${ax^2 + bx + c = 0}$; then which of the following is not correct? A. a is nonzero - I ruled out this because if was 0 ...
3
votes
0answers
72 views

Fibonacci Quadratic Residue

After some research I have came up with a conjecture on fibonacci quadtratic residue: ...
0
votes
0answers
44 views

About a Variant of Ulam Spiral

Here I read about a variant on the Ulam spiral: [A] structure may be seen when composite numbers are also included in the Ulam spiral. [...] Using the size of the dot representing an integer ...
2
votes
1answer
41 views

What is the greek symbol that represents the ratio of the length and with in a rectangle.

I understand that there is pi, but I was provoked by a question that asked this, "I have a rectangle and if I cut a square off of it it produces the same type of rectangle. I know that the width is 1 ...
0
votes
0answers
34 views

Calculating $\arg\min_x (1-\Phi(x;\mu_1,\sigma_1^2)+\Phi(x;\mu_2,\sigma_2^2))$

I would like to find $x$ satisfying the following expression: $$\arg \min_x R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2)$$ where $$R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) ...
0
votes
0answers
18 views

Ellipse equation. What does it need to be in order for $b > a$?

We have the quadratic equation: $$ax^2 + bx + cy^2 + dy + e$$ $a$ and $c$ are both negative or both positive. How can I, by looking at that only, determine whether $b$ (the length of the semi-minor ...
2
votes
4answers
54 views

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take?

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take? This is what I have done: Let $y = 3^x$ $$9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$$ $$\implies9y^2 + (t^2 - 4t - 2)y + ...
0
votes
2answers
130 views

How factor with square root

I have the following equation that I'm trying to factor, but I'm stuck at the end. $$\frac{zx^{-4}\sqrt{x}(yz^4)^3}{z^7xy}$$ $$\frac{\frac{1}{x^4}\sqrt{x}(yz^4)^3}{z^6xy}$$ ...
1
vote
3answers
28 views

Need help with basic factoring equation

I'm just trying to brush up on my factoring of quadratic equations. $$\frac1{x+3} + \frac1{x^2 + 5x +6}$$ $$\frac1{x+3} + \frac1{(x+2)(x+3)}$$ $$\frac{(x+2)(x+3) + (x+3)}{(x+2)(x+2)(x+3)}$$ Then ...
3
votes
1answer
78 views

Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
3
votes
0answers
133 views

Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real root…

Problem : Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real roots belonging to the interval $(1,2) $ then the minimum possible values of a is ...
-1
votes
2answers
29 views

Quadratic equation form?

Suppose we know that the sum of two positive numbers is $2k$ and their product is $m$ then which of the following will be its quadratic equation and why? 1) $x^2$+ $(2k)x$+ $m$= $0$ 2) $x^2$- ...
0
votes
4answers
71 views

When do you use the quadratic formula?

I am revising for a mathematics exam and am looking over simultaneous equations. I was curious as to when I use the quadratic formula and when I don't? I realize there are multiple ways to solve a ...
1
vote
1answer
57 views

how to solve these two quadratic equations

Can someone help me find the solution for these two quadratic equations ? $ 2(z^2) \ - \ 3.023bz \ + \ 0.115(b^2) \ + \ 2.0814b \ + \ 0.142z \ - \ 0.5856 \ = \ 0 $ $ 6.0828(z^2) \ + \ 2.0414bz \ + \ ...
0
votes
1answer
20 views

What is the minimum possible $ non $ integral value of a

Let a A subscript(m) (m=1,2,3,....p) be the possible integral values of a for which the graphs of $ f(x)=ax^2+2bx+b $ and $g(x)=5x^2-3bx-a$ meets at some point for all real values of b. 1) What is ...
3
votes
1answer
187 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
2
votes
1answer
29 views

A question on multinomial theorem using binomial theorem

$(3x^2+2x+c)^{12}=\sum A_r x^r$ and $\frac{A_{19}}{A_5}=\frac 1 {2^7}$ Find $c$. I really have no idea what to do with this. This was on a test. I have studied only binomial theorem. So, please ...
0
votes
1answer
50 views

Determining quadratic function of this word problem

I have this word problem in my homework: ...
1
vote
1answer
70 views

A Cubic Equation

$2x^3+ax^2+bx+4=0$, $(a,b \in R^+)$ has three real roots. Then : A. $a\geqslant 4.2^{\frac 1 3}$ B. $a\geqslant 1.2^{\frac 1 3}$ C. $a\geqslant 6.2^{\frac 1 3}$ D. $a\geqslant 2.2^{\frac 1 3}$ ...
1
vote
1answer
31 views

Terminology of “linear”, “quadratic”, etc. for multi-input functions

It is my understanding that, according to typical math terminology: The function $f(x, y) = x + y$ is "linear". Specifically, it's linear in both $x$ and $y$, but this is understood implicitly. ...
1
vote
2answers
143 views

number of integral values for which $x^2+19x+92$ is a perfect square.

number of integral values of x for which $x^2+19x+92$ is a perfect square=? I have no idea how to do this. Please help.
1
vote
0answers
21 views

Variable quadratic functions

I have some doubts in proving the following:- C is the curve $$y = \frac 1 {k+1} [2x^2 + (k + 7)x + 4]$$, where k is a real number not equal to -1. Show that C always passes through two fixed points ...
0
votes
2answers
133 views

How would I solve the quadratic $x^2+3x-70=0$?

How would I solve the following quadratic equation $$x^2+3x-70=0 $$ This is my attempt below $$(x-7x) (x+10x)=0 $$ $$ x-7x=0 \implies -6x=0 \implies x=6$$ $$x+10x=0 \implies 11x=0 \implies ...
0
votes
1answer
72 views

How to interpret coefficients in polynomial regression?

I am working on my thesis (study) about poverty incidence rate and its socio-economic factors using second-order polynomial regression without interaction. The final model in my study is ...
0
votes
1answer
21 views

Verification of solutions to some polynomial prob/

$\boxed{\text{Problem 1}}$ Find the other solution of the equation $(1+\sqrt3)x^2-(5-\sqrt3)x+6-6\sqrt3=0$ given that $2$ is a solution Ma solution: $x_1\cdot x_2=\tfrac ca$ therefore letting ...
3
votes
2answers
37 views

prove for p(x) which is a quadratic polynomial

$p(x)$ is a quadratic polynomial . Prove that any given number for $a$ with one exception , we can find a number $b$ such that $p(a)=p(b)$ and $a$ is not equal to $b$.