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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3
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2answers
281 views

How do you find the vertex of a (Bézier) quadratic curve?

Before I elaborate, I do not mean a quadratic function! I mean a quadratic curve as seen here. With these curves, you are given 3 points: the starting point, the control point, and the ending point. I ...
4
votes
3answers
92 views

Find the number of values of $a$?

Consider a quadratic equation; $$ x^2 + 7x – 14(a^2 + 1) = 0,$$ … (where $a$ is an integer) For how many different value of $a$, the equation will have at least one integer root? I found out its ...
7
votes
6answers
358 views

Solve $5a^2 - 4ab - b^2 + 9 = 0$, $ - 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0$

Solve $\left\{\begin{matrix} 5a^2 - 4ab - b^2 + 9 = 0\\ - 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0. \end{matrix}\right.$ I know that we can use quadratic equation twice, but then we'll get some ...
0
votes
1answer
210 views

An airplane makes a 990 km flight with a tailwind and returns, flying into the same wind.

An airplane makes a 990 km flight with a tailwind and returns, flying into the same wind.The total flying time is 3 hrs 20 mins and the airplanes speed in still air is 600 km/h what is the speed of ...
2
votes
6answers
262 views

Solving for x with exponents (algebra)

So I am trying to help a friend do her homework and I am a bit stuck. $$8x+3 = 3x^2$$ I can look at this and see that the answer is $3$, but I am having a hard time remembering how to solve for $x$ ...
4
votes
2answers
128 views

Finding the Extrema of a Function (without differetiation)

$$ (t^2-t+1)/(t^2+t+1) $$ prove that the function is upper bounded by 3 and lower bounded by 1/3 without differentiation
1
vote
1answer
78 views

How to find sum of quadratic

I got this quadratic function from physics that I need to find the sum of each term, up to whatever point. Written thusly: $$ \sum_{n=1}^{t}4.945n^2$$ And is there someway to quickly figure this ...
2
votes
3answers
4k views

Taking the square roots in inequalities

I have a question regarding taking square roots in inequalities. I have a problem asking: Suppose $3x^2+bx+7>0$ for every real number x. Show that $|b|<2\sqrt{21}$. In an earlier question it ...
1
vote
4answers
306 views

Is it possible to take the absolute value of both sides of an equation?

I have a problem that says: Suppose $3x^2+bx+7 > 0$ for every number $x$, Show that $|b|<2\sqrt21$. Since the quadratic is greater than 0, I assume that there are no real solutions since $y = ...
4
votes
4answers
420 views

Quadratic function concepts

My teacher was explaining quadratics in my class and it was a little bit unclear to me. The problem was Suppose $at^2 + 5t + 4 > 0$, show that $a > 25/16$ . My teacher said that there are ...
1
vote
1answer
3k views

Completing the square with negative x coefficients

I know how to complete the square with positive $x$ coefficients but how do you complete the square with negative $x$ coefficients? For example: \begin{align*} f(x) & = x^2 + 6x + 11 \\ & = ...
1
vote
1answer
63 views

What technique reduces factorable ax^2+bx+c=0 to factorable where a=1

The normal way to factor ax^2+bx+c=0 is to look for t,u,v,w such that: (tx+u)(vx+w) = 0 so that tv=a, uw=c, and uv+wt=b. This can be tricky, since there can be several possibilities for t,u,v,w. ...
3
votes
2answers
251 views

Quadratic equation to calculate a temperature from resistance

I'm trying to implement an electronic temperature sensor that gives a resistance value. The sensor is a Honeywell TD4. In the datasheet, they give a table of values : -40ºC => 1584Ω ±12Ω -30ºC => ...
2
votes
3answers
648 views

How was the quadratic formula found and proven? [duplicate]

Possible Duplicate: Why can ALL quadratic equations be solved by the quadratic formula? History of Quadratic Formula How was the quadratic formula $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ found ...
3
votes
4answers
238 views

Where is the order of the variables inside the parentheses coming from?

I'm reading Sawyer's Prelude to Mathematics, here: I can't understand what's the meaning and application of "condition" here. Also when he gives the example on the cubic equation, stating that ...
1
vote
2answers
1k views

Mathematics behind intersection points of two lines using quadratic equation

This is the question I am trying to solve. I do not need any code examples just help on mathematics. Suppose two line segments intersect. The two endpoints for the first line segment are $(x_1, ...
5
votes
5answers
4k views

Why a quadratic equations always equals zero?

On evaluating quadratic equations, It always equals zero: $$ax^2+bx+c=0$$ Why zero? Is it possible to use other number for another purpose?
1
vote
5answers
1k views

On the number of possible solutions for a quadratic equation.

Solving a quadratic equation will yield two roots: $$\frac{-\sqrt{b^2-4 a c}-b} {2 a}$$ and: $$\frac{\sqrt{b^2-4a c}-b}{2 a}$$ And I've been taught to answer it like: $$\frac{\pm\sqrt{b^2-4a c}- ...
1
vote
2answers
93 views

Let, both $a$ and $b$ belong to the set {1,2,3,4}. What is the number of equations of the form $ax^2+bx+1=0$ which have real roots.

Let, both $a$ and $b$ belong to the set {1,2,3,4}. What is the number of equations of the form $ax^2+bx+1=0$ which have real roots. for real roots, $a \gt 0$, $b^2-4{a}{c} \ge 0$ Here we have $c=1$, ...
1
vote
3answers
206 views

How do I solve a Continued Fraction of solution to quadratic equation?

I know that it is possible to make a CF (continued fraction) for every number that is a solution of a quadratic equation but I don't know how. The number I'd like to write as a CF is: $$\frac{1 - ...
0
votes
0answers
48 views

In a quadratic extension, does the orthogonal complement of trace have a nice name?

The trace of an element in a quadratic extension is the sum of it with its conjugate. Is there a name for the difference between it and its conjugate?
1
vote
1answer
591 views

Getting square root of negative in completing the square problem

I try to solve the equation $f(x) = 7x - 11 - 2x^2 = 0$ for $x$, but run into troubles. I've gone through it over and over again as well as similar problems, but can't find what I'm doing wrong. ...
3
votes
3answers
318 views

Does $x^2 \equiv 211\pmod{ 159}$ have a solution?

Note that 159=3*53. The answer to this question is yes. I managed to find two of the solutions. They are $x=23,136$ but there are two more. The main question that I have is whether there is an easier ...
1
vote
2answers
49 views

How do I transform the equation based on the condition?

If $q$ and $w$ are the roots of the equation $$2x^2-px+7=0$$ Then $q/w$ is a root of ? P.s:- It is an another question of How do I transform the equation based on this condition?
2
votes
1answer
44 views

How do I transform the equation based on this condition?

If a and b are the roots of the equation $$2x^2-px+7=0$$ Then a-b is a root of ?
2
votes
2answers
128 views

How do I proceed with these quadratic equations?

The question is $$ax^2 + bx + c=0 $$ and $$cx^2+bx+a=0$$ have a common root, if $b≠ a+c$, then what is $$a^3+b^3+c^3$$
0
votes
2answers
350 views

Definition of quadratic equation?

What is a quadratic equation and what is its simplified and cannonic form?
1
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3answers
410 views

Changing from quadratic formula to standard form.

The graph of a quadratic function has $x$-intercepts $-1$ and $3$ and a range consisting of all numbers less than or equal to $4$. Determine an expression for the function. This is my problem. I ...
2
votes
2answers
440 views

What are the benefits of using $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ to solve quadratic equations?

When I was in high school, they taught me to solve quadratic equations with this formula: $$x=\frac{\sqrt{4 \text{ac}+b^2}-b}{2 a}$$ EDIT: The original formula is this one: $x = \dfrac{-b \pm ...
1
vote
3answers
737 views

Quadratic equation with absolute value

Prepping for the GMAT, I came across the following question: What is the product of all solutions of: $$x^2 - 4x + 6 = 3 - |x - 1|?$$ First, I set up two equations, ie: $$x^2 - 4x + 6 ...
3
votes
5answers
227 views

Proving Quadratic Formula

purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become ...
5
votes
2answers
888 views

Solution af a system of 2 quadratic equations

I have a system of two quadratic equations with unknowns $x$ and $y$: $$a_{1 1} x y + a_{1 2} x^2 + a_{1 3} y^2 + a_{1 4} x + a_{1 5} y + a_{1 6} = 0,\\ a_{2 1} x y + a_{2 2} x^2 + a_{2 3} y^2 + a_{2 ...
1
vote
3answers
117 views

Help with radical equation

Please, help me to solve this equation. No advanced math should be needed. $$ 3x^2 - 4x + \sqrt{3x^2 - 4x - 6} = 18 $$ I'm clueless. It should be simple.
0
votes
2answers
225 views

Solving a set of “circular” quadratic equations

$x_a'$ and $y_a'$ are unknown. What's the simplest way to solve it? Every time I tried, it grew into tremendous size or was unable to think out in reasonable amount of time due to it's complexity. ...
1
vote
2answers
132 views

Quadratic System of Equations

I'm trying to define a quadratic that can pass through any 3 points. I've obviously done something wrong but can't figure out where. Any help would be appreciated. $$ ax_1^2 + bx_1 + c = y_1 $$ $$ ...
17
votes
3answers
397 views

What would be the value of $\sum\limits_{n=0}^\infty \frac{1}{an^2+bn+c}$

I would like to evaluate the sum $$\sum_{n=0}^\infty \frac{1}{an^2+bn+c}$$ Here is my attempt: Letting $$f(z)=\frac{1}{az^2+bz+c}$$ The poles of $f(z)$ are located at $$z_0 = ...
3
votes
1answer
446 views

Given a cubic function, and its quadratic derivative- can I recover the cubic from quadratic?

Background: I'm trying to learn how to work with cubic and quadratic bezier splines for various drawing libraries, and working through how to approximate a cubic spline with a quadratic spline. It's ...
25
votes
4answers
4k views

Is it possible for a quadratic equation to have one rational root and one irrational root?

Is it possible for a quadratic equation to have one rational root and one irrational root? Yes, a pretty straightforward question. Is it possible?
2
votes
4answers
178 views

Solving a quadratic Inequality

My question is: Solve $$9x-14-x^2>0$$ My answer is: $2 < x < 7$ Though I know my answer is right, I want to know in what ways I can solve it and how it can be graphically represented. ...
2
votes
5answers
301 views

$3x^3 = 24$ quadratic equation

Completing the square I know by factoring $$x^3 - 8 = 0\\ x-2 = 0$$ that one of the solutions is 2. but the other solutions is $1 ± i \sqrt 3$. Can someone explain to me how to get that?
1
vote
3answers
106 views

Does a quadratic necessarily have a root in this interval?

If F(x) is the quadratic $ax^2+bx+c$ with $ac>0$ $b^2-4ac>0$, it is true that within the interval $[-\frac{b}{a},+\frac{b}{a}]$ there exists a point $x$ where $F(x)=0$. I was told this earlier ...
0
votes
1answer
189 views

Finding a Parabola from its height and second y-intercept

I would like to know how to find the quadratic equation for a parabola that opens down and intersects the origin along with the vertex being in the first quadrant given the maximum of the parabola and ...
3
votes
1answer
255 views

Recursive sequence and a quadratic equation related inequality proof

I am trying to show that if a sequence of number $x_{n}$ is defined by $x_1 = h$, $x_{n+1}=x_n^2 + k$, where $0<k<\frac{1}{4}$ and $h$ lies between the roots $a$ and $b$ of the equation $$x^2 -x ...
1
vote
0answers
65 views

Having trouble with finding a Quadratic Expression

I am having trouble trying to work out a quadratic expression for uni, being a external student its hard to find help. My problem is Perimeter = 1000m Part 1 Solve L in terms of w in regard to the ...
3
votes
3answers
337 views

Unfoiling quadratic equation

How can I convert $ax^2 + bx + c = 0$ to a FOIL-style $(x + d)(x - e) = 0$ equation? I have an equation in a computer program that I'm currently solving with the standard $\frac{-b \pm \sqrt{b^2 - ...
5
votes
4answers
570 views

find the least a, for which two equations have a common root

Could you help me out please. I have two equations: $2x^2-3x+1=0 $ and $ 2x^2-(a+3)x+3a=0$ I need to find the least $a$ for which these two equations have a common root. At a first glance I thought ...
2
votes
3answers
117 views

How to find orignal equations of type $y=ax^2+bx+c$. given 3 coordinate points?

Ok, simple question, having trouble understanding this in school. So given a set of 3 points (xy-plane), such as (40,30) (60,28) (20,25) i have to find the equation of the parabola. I ...
4
votes
2answers
107 views

Getting a standard form quadratic from a set of points ($3$ points)

I came home from school today, pulled out my homework, now I'm stumped. I don't want the answer, I just want to know how to do it. Here is the question that I'm reading: Determine a quadratic ...
3
votes
1answer
166 views

Close Packing of Ellipsoids

How can the packing density of a set of congruent ellipsoids be calculated? I'm dealing with prolate spheroids so technically I do not need the general answer for ellipsoids, but my abstract mind ...
4
votes
2answers
577 views

Determine if equation will generate perfect squares

Given the following quadratic equations: $4n^2 + 128n - 131$ $4n^2 + 16n - 11$ $4n^2 + 24n - 3$ Is it possible to determine how many values of n will generate a perfect square? Or better yet, is ...