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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4
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1answer
170 views
+50

Can we use matrix to solve this inequality?

Let $$f(x)=\begin{cases} 1&0\le x\le 1\\ 0&\rm{others} \end{cases}$$ Let $x_{i},a_{i}(i=1,2,\cdots,n)$ be positive real numbers, show that: ...
1
vote
2answers
35 views

Quadratic formula errors

I'm clearly making a silly mistake here, but I can't see it. EDIT: I missed brackets when typing out the expression to calculate. Apologies for timewasting. I have the equation $(2x + 3)(5x + 1)=0$. ...
2
votes
4answers
49 views

Show that no line with a y-int of 10 will ever be tangential to the curve with $y=3x^2+7x-2$

Show that no line with a y-int of 10 will ever be tangential to the curve with $y=3x^2+7x-2$. Having trouble in showing this. So far these are my process. Let line be $y=mx+10$ $mx+10 = 3x^2+7x-2$ ...
-2
votes
2answers
31 views

Figuring $x$ and $y$ from two linear equations

I have a mini exam in a month to study for and I'm looking at systems of equations at the moment. I have this question to look at right now: Find $x$ and $y:$ $x-5y+4=1$ $\dfrac{x+1}{2}=y^2$ Now ...
3
votes
3answers
65 views

Need help solving $x^4-3x^3-11x^2+3x+10=0$

Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ...
2
votes
2answers
761 views

Optimization: maximum area of a triangle under a parabola

Optimization: maximum area of a triangle in a parabola Inside a curve ($x^2-25$ - Parabola) a triangle is drawn with A as the vertex at the origin and the line joining points B and C lie on the ...
2
votes
1answer
25 views

What is the solution for $y(t)=e^{-\frac{t}{\tau y(t)}}$?

A simple quadratic flow model leads to the following apparently simple equation $$y(t)=e^{-\frac{t}{\tau y(t)}}$$ where the flow, $y$ is a function of time, $t$ and $\tau $ is a constant. But is ...
0
votes
0answers
34 views

Testing if a Point is inside an oriented ellipsoid

So my goal is to do a search for the points that are inside an ellipsoid. for that i need to test if each point is inside. I need to be able to orient my ellipsoid, so i looked up how to write a ...
0
votes
1answer
24 views

If $x^4 + 3\cos(ax^2 + bx +c) = 2(x^2-2) $ has two solutions with $a,b,c \in (2,5)$, then find the maximum value of $\frac{ac}{b^2} $

The answer given is 1. i tried like this $3\cos(ax^2 + bx +c) = -x^4 +2x^2-4 = -(x^2 -1)^2 -3 $. The maximum value of $-x^4 +2x^2-4$ is $-3$ so $3\cos(ax^2 + bx +c) =-3$ and the two values of x are ...
0
votes
2answers
48 views

Can't find my mistake

I'm trying to find the sum of the reciprocal numbers of squares of quadratic equation:$3x^2-14x+6=0$, I managed to find the answer by calculating the roots, and summing their reciprocal. However ...
0
votes
1answer
56 views

Equations $ax^2+btx+c=0, bx^2+ctx+a=0$ and $cx^2+atx+b=0$

Find different real numbers $a,b,c,t$ for which the following conditions: 1) the equation $ax^2+btx+c=0$ has real roots $x_1,x_2$; 2) the equation $bx^2+ctx+a=0$ has real roots $x_2,x_3$; ...
5
votes
3answers
474 views

Solving Radical Equations $x-7= \sqrt{x-5}$

This the Pre-Calculus Problem: $x-7= \sqrt{x-5}$ So far I did it like this and I'm not understanding If I did it wrong. $(x-7)^2=\sqrt{x-5}^2$ - The Square root would cancel, leaving: ...
1
vote
2answers
59 views

possible real solutions of the equations

What are the possible real solutions of the equations $$1000=v_1^2+4v_2^2,100=v_1+4v_2$$ Its a physics question but I thought its not necessary to post here . Thank you.
3
votes
1answer
50 views

Roots of the equation $x^2+1=0$ in $\Bbb Z/p^{n}\Bbb Z$

Let $p$ be an odd prime number and $n$ be a positive integer. I want to consider roots of the equation $x^{2}+1=0$ in the ring $\Bbb Z/p^{n}\Bbb Z$. Suppose $n=1$. Find a condition on $p$ such ...
-1
votes
2answers
21 views

Determine a, b & c in a (possible) Quadratic equation [closed]

given f(x)=x: 1 - f(5)=2 2 - f(3)=3 how to determine the coefficients a, b & c in the polynomial equation like: ...
0
votes
3answers
99 views

max of $e$ with $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$ [closed]

Given that a,b,c,d,e are real number such that: $\begin{cases} a+b+c+d+e=8\\ a^2+b^2+c^2+d^2+e^2=16 \end{cases}$ determine the maximun value of $e$. I started like that : ...
0
votes
1answer
52 views
0
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1answer
40 views

Are there any tricks for simultaneous equations I should be aware of?

I'm at the end of a difficult logarithms question and have ascertained the linear equations I need in order to establish x and y as the questions asks of me. The equations are: $x - 5y + 4 = 1$ ...
0
votes
1answer
27 views

Find set of values for $k$ with two distinct roots?

Find the set of values for $k$ for which the line $y=2x-k$ meets the curve $y=x^2+kx-2$ at two distinct points. I have started my equation like this: $$x^2+kx-2=2x-k$$ but I need to put it in the ...
-1
votes
2answers
35 views

$100 + [110/(1+r)] = [1/ (1+r)] + [(232 /(1+r)^2 ]$

Need to learn how to solve this: $100 + \frac{110}{1 + r} = \frac{1}{1 + r} + \frac{232}{(1 + r)^{2}}$. Checked this site got to the 3rd line and am completely lost. Can someone help me solve for r ...
0
votes
0answers
13 views

Indefinite Boolean Quadratic Programming: number of minima

The Boolean Quadratic Programming problem is defined as: $\min_{x} f(x) = x^TQx + c^Tx$ s.t. $ x \in \{0,1\}^n$ It is a well-studied NP-Hard problem with many approximation algorithms proposed. I ...
-1
votes
1answer
33 views

How to expand a equation into a quadratic equation? [closed]

I need a explanation and steps how I can expand this equation: $$(x−p)^2+(mx+c−q)^2=r$$ into this one: $$(m^2+1)x^2+2(mc−mq−p)x+(q^2−r+p^2−2cq+c^2)=0$$ Thank you for your time.
0
votes
1answer
634 views

How do I prove a quadratic is always positive or negative for x?

I looked this up and seen something that was beyond my A-Level Maths course. In class we are doing the discriminant and sketching quadratic graphs, so it is nothing advanced. My teacher completed ...
0
votes
1answer
473 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 ...
2
votes
2answers
29 views

Find the set of values of x for which $\frac{x+1}{2x-3}<\frac{1}{x-3}$

Here's what I've done: $\frac{x+1}{2x-3}<\frac{1}{x-3}$ $x+1<\frac{2x-3}{x-3}$ $(x+1)(x-3)<2x-3$ $x^2-2x-3<2x-3$ $x^2-4x<0$ $x(x-4)<0$ $0<x<4$ However this clearly ...
3
votes
1answer
41 views

Solution to a simple system of quadratic equations

I am hoping to find a closed-form solution to the following system of $n$ quadratic equations: $$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$ for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial ...
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 ...
-6
votes
1answer
58 views

Range of the expression $\frac{9 \cdot 3^{2x}+6\cdot 3^{x}+4}{9 \cdot 3^{2x}-6 \cdot3^{x}+4}$ [closed]

Given that, for all $x \in \mathbb{R}$ the expression $\frac{x^2-2x+4}{x^2+2x+4}$ lies between $1/3$ and $3$ the values between which the expression $\frac{9 \cdot 3^{2x}+6\cdot 3^{x}+4}{9 \cdot ...
0
votes
0answers
20 views

Involves finding minimum length as x varies

Ok in a question where you have been given coordinates for a and b (-3,5),(x,3x+9) respectively and AB^2 is given to be 10X^2+30x+25 for this 5 is taken out as a factor so you're left with ...
1
vote
0answers
28 views

Am I finding this $x$-value correctly?

If the flight path of a cricket ball is given by: $$y = \frac{1}{3}x - \frac{1}{60}x^2$$ And a fielder standing originally at $(10, 0)$ catches the ball when it is $1.5$ units above the ground, to ...
5
votes
2answers
6k views

Finding the discriminant and roots of a polynomial

How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where $f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ and here $\Delta$ is the ...
2
votes
1answer
39 views

$f(x)$ is a quadratic polynomial with leading coefficient $1$, $|f(x)| \leq 8 \: \forall \: x \in [1,9]$ find $f(x)$

$f(x)$ is a polynomial of the form ($b,c$ are real numbers) $$f(x) = x^2+bx+c$$ such that $|f(x)| \leq 8 \: \forall \: x \in [1,9]$. Find all $f(x)$ satisfying the given condition. I found ...
0
votes
2answers
66 views

Quadratic Equation Roots Prove

I have a question in my textbook from chapter of quadratic equations from exercise of sum of roots and product of roots that; Prove that the equation $$ a x^2 + b x + c = 0, \quad a > 0 $$ has ...
1
vote
4answers
81 views

Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

I have a question. Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$ I don't know how to proceed. ...
0
votes
3answers
53 views

finding rational roots

Consider the integral expression in $x$ $$P=x^3+x^2+ax+1,$$ where $a$ is a rational number. At $a= ?$ the value of $P$ is a rational number for any $x$ which satisfies the equation $x^2+2x−2=0$, and ...
0
votes
1answer
18 views

Total number of integral solutions for the given second degree equation!

First, the problem statement : "Consider the equation $x^{2}+y^{2}-3z^{2}-3t^{2}=0$. The total number of integral solutions of this equation in the range of the first 10000 numbers, i.e., $1\leq ...
0
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2answers
75 views

Soft Question: Weblinks to pages with explanation on quadratics.

I recently placed a question based on quadratics and received a few valuable answers. One of them was a comment in an answer with a link in it which I found useful. But unfortunately the webpage (of ...
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votes
2answers
48 views

Does a square root come out plus/minus even if there is a negative sign outside?

For example: $-\sqrt{100x^{20}y^{10}}$. Would that give $\pm10x^{10}y^5$ or just $-10x^{10}y^5$?
1
vote
1answer
44 views

Fit a Quadratic Curve to Data

I have some data and I want to fit a quadratic curve for my data But I don't know that how to it do? My data : $x,y = 100,45;$ $x_1,y_1= 101, 50$; $x_2,y_3=99,35$; $\ldots$ For instance this ...
0
votes
0answers
20 views

How to use the complete the square method for a given function.

I am given a function $u(y_1, y_2, y_3)= (y_1)^2+(y_2)^2+(y_3)^2-2y_1+2$ for $y$ on the boundary of the ball $B(x,2)$ and it ends up that this function is equal to $u(y) = |y-x|^2 +1$, $y$ and $x$ are ...
0
votes
1answer
37 views

If $ a(12a + 5b +2c) > 0 $ , then prove that the real roots of equation $ ax^2+ bx +c =0$ are less than 2

I know that to prove both roots less than 2 ,i have to prove $ 4a+ 2b +c > 0$ and $ -b/2a < 2 $ . Here i have no idea how to proceed.
0
votes
0answers
38 views

How can you solve a polynomial with a power of 1.99?

$y_2 = -.0251256t^{1.99} + (v_{y})t + y_1$ How would I solve for $t$? Ideally I would approximate as a quadratic, but the error becomes too high.
4
votes
10answers
133 views

Factor $6x^2​ −7x−5=0$

I'm trying to factor $$6x^2​ −7x−5=0$$ but I have no clue about how to do it. I would be able to factor this: $$x^2-14x+40=0$$ $$a+b=-14$$ $$ab=40$$ But $6x^2​ −7x−5=0$ looks like it's not ...
3
votes
1answer
34 views

When are we able to find a quadratic with roots that are a function of another quadratic?

Motivation: Given the roots of the quadratic $2x^2+6x+7=0$ find a quadratic with roots $\alpha^2-1$ and $\beta^2-1$ I was able to solve this problem in two ways: Method 1: Sum of the roots ...
2
votes
2answers
75 views

If $\alpha_1,\alpha_2,\ldots,\alpha_n$ be the roots of the equation $x^n+1$

then $(1-\alpha_1)(1-\alpha_2)\ldots(1-\alpha_n)$ equals to ? I think here we need the info of whether $n$ is even or odd else how will we say whether by vieta's formula what is the value of ...
0
votes
2answers
12 views

How to find the solutions for the quadratic equation for conic sections $\epsilon \in (0,1)$

Going from this definition of the conic section: $\epsilon |Pl| =|PB|$, you get the following equation for the intersection with the $x$-axis: $y^2 = (\epsilon ^2-1)x^2+(B-\epsilon ^2L)2x+\epsilon ...
1
vote
1answer
44 views

Solving octic equation using quadratic formula

According to the wikipedia article on octic equations, octic equations of the form $ax^8 \pm bx^4 \pm c = 0$ can be solved using the quadratic formula. How might one actually do this?
2
votes
0answers
42 views

question on quadratic expansion [closed]

I have been trying to solve this question, but no luck so far, any help would be appreciated. Let $a,b,c > 0$ be such that $a^2 + b^2 -2bc =100, \ 2ab -c^2 = 100$. Then the value of ...
3
votes
4answers
67 views

$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$ - I keep getting imaginary numbers

$$\sqrt{x+938^2} - 938 + \sqrt{x + 140^2} - 140 = 38$$ My attempt $\sqrt{x+938^2} + \sqrt{x + 140^2} = 1116$ $(\sqrt{x+938^2} + \sqrt{x + 140^2})^2 = (1116)^2$ $x+938^2 + ...
1
vote
1answer
19 views

find the equation of the diameter which passes through the origin.

I am given the equation of the circle $x^2+y^2−4x+6y=14$, and I am told to find the equation of the diameter which passes through the origin. However, I am unsure as to how to do this.