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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0
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4answers
34 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 $3x^2+4x-4=0$.
7
votes
3answers
114 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
42 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 ...
3
votes
2answers
70 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
24 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
1answer
42 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
30 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
33 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
24 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
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 ...
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0answers
10 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
28 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
47 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
96 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
11 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
27 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
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2answers
26 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
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2answers
33 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
64 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
59 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
26 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
36 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
38 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
41 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
14 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
134 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
28 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
56 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
22 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
66 views

Fibonacci Quadratic Residue

After some research I have came up with a conjecture on fibonacci quadtratic residue: ...
0
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0answers
38 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
37 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
25 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
17 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
51 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
45 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
26 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
75 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 ...
2
votes
0answers
89 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
27 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
68 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
53 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
19 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
169 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
23 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 ...