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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4answers
39 views

Finding the three unknowns

Can someone show me the steps to finding the three unknowns of these two equations. $$-a-bx+cx^2 = x^2+2x+1$$ The answers are $a=\ ...\ $, $b=\ ...\ $, and $c=\ ...$ , but I can't see how they ...
0
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1answer
45 views

Homework: canonical form of quadratic form

X=(x,y,z) Q(X) = $x^2 + 4xy + 6xz + 3y^2 +8yz +5z^2 $ I got by using completing the square method: Q(X) = $(x+2y+3z)^2 - (y+2z)^2$ so as I learned now I do: $u = x+2y+3z$ $v = y+2z$ $w = 0 $ ...
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1answer
17 views

system of equations solving with only that information

Hi would would I go around to solve the following, there is no other information stat is given other than the fact that i have already expanded this from this $(25-y)(x+8)=523$ $25x-8y=323$
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1answer
53 views

Calculate the volume of water in glass over time.

For A) I found that volume should be defined by But I got no idea what to do in b) and c)
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0answers
51 views

Showing DO NOT exist GCD of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$.

Showing DO NOT exist gcd of $6$ and $2+2 \sqrt{-5}$ in $\Bbb Z[\sqrt{-5}]$. I tried it. Suppose $d$ is GCD of $6$ and $2+2 \sqrt(-5)$. then there exist $x,y \in \Bbb ...
2
votes
1answer
65 views

Expanding Square Roots, Why No Negative?

I haven't thought through algebra in a while and the last explanation I received of this seemed arbitrary. I hope I can get some clarification here. I understand that $\sqrt{+a} = \pm b$. Here's ...
1
vote
1answer
47 views

Quadratic inequality with parameter

Hi I've got this inequality with parameter $a\in R$ $\frac{x+a}{x}\le x+2$ I've solved it but I've got different results than book. I've done it by dividing it into 2 cases. 1. x<0 2. x>0 and then ...
4
votes
7answers
89 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 ...
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1answer
28 views

Proof of axis of symmetry [duplicate]

How do you prove -b/2a the Axis of symmetry equation using the Quadratic formula?
-1
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2answers
39 views

Proof of axis of symmetry equation [closed]

Because quadratic functions are symmetrical how do you prove the axis of symmetry equation. $x=(-b/(2a))$
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1answer
77 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
37 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 ...
1
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2answers
46 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) ...
6
votes
4answers
125 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 ...
0
votes
5answers
48 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
141 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
216 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 ...
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4answers
36 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 $3x^2+4x-4=0$.
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3answers
116 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
43 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
72 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
43 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. $q(10)$ ...
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1answer
32 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
35 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
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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 ...
0
votes
0answers
12 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
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1answer
46 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
50 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
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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
103 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
31 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
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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
60 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
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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
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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
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
48 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" ...
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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
140 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
37 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?
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2answers
59 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 ...