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 $y=a(x+b)(x+c)$.

learn more… | top users | synonyms (1)

0
votes
2answers
49 views

Quadratic Equation to prove $ax^2+bx+c=0$

"Prove that there is one and only quadratic equation for which the sum of the roots is $3$ and the cubed of the roots is $63$" I'm practicing for the Maths Olympiad. I'm a high school student and ...
2
votes
2answers
35 views

Do i need to always take out factor of -1 when solving a quadratic in form $ax^2 + bx + c$ when $a$ is negative?

In my textbook I am advised to always take out the factor of -1 if the quadratic is in the form of $ax^2 + bx + c$ when $a$ is negative. For example: $-4x^2 + 4x + 3$ My question is, is ...
2
votes
3answers
38 views

Solve an equation involving $r$ in the denominator

Sorry if the title is wrong, I am not sure what to call this. It's been a really long time since I've taken math, and I can't remember for the life of me how to solve this, nor can I figure out how to ...
0
votes
0answers
12 views

Finding points on a parabolic arc [duplicate]

So, I'm a game programmer and not really strong in math. I'm making a 3D game (I use an engine that does most of the math work for me in various areas but not this one) where I can throw a grenade. I ...
0
votes
1answer
27 views

Possible even integer values of $y$ under a system of quadratic and linear equations

If $$25(9x^2+y^2)+9z^2-15(5xy+yz+3zx)=0$$ and $$x+y+z=18$$ then possible even integral values of ($y$) less than $15$ are: ? My attempt: I manipulated the given equation in this form ...
0
votes
1answer
18 views

Intuitive proof for $\forall n\in\mathbb{N}\ \ \forall k\in\mathbb{N}, k<n \ \colon \ \ k(k+1)\not\in [n^2, n(n+1)]$

I'm trying to find an intuitive and quick proof for the statement in the question. In my exercise book it's just stated that this is so without proving it. I'm trying to find a quick proof which ...
0
votes
1answer
43 views

Solve for k when the equation has equal roots

UPDATE: Solved thanks to turkyhundt and jimbo The mathematical question is as follows: Calculate the value of k for which 2x^2 + 4x - k = 0 has equal roots. My working solves it to equal -2, ...
-1
votes
2answers
58 views

Equation of parabola with given $y$-intercept and roots

I have two questions involving quadratics. Next to a diagram of a parabola with a ma point with the y intercept of $(0,p)$ and roots $(-1,0)$ and $(p,0)$ it says: 3a) Show that the equation of the ...
0
votes
1answer
52 views

Can the system $x+y=3$, $2x^2 + y^2 = 5$ be solved using matrices?

$$ x+y=3 $$ $$ 2x^2 + y^2 = 5 $$ I solved it by substituting $x = 3- y $ $2(3-y)^2 + y^2 =5 $ therefore, $ y= 2+\frac{i}{\sqrt3} $, $y= 2-\frac{i}{\sqrt3}$ However, I want to know that can I ...
-3
votes
1answer
31 views

Matrix form of a quadratic function with 3 variables

$$f(x,y,z)=x-2y+3z^2$$ When given this quadratic function, how would I write it in a matrix form? I have been taught that any quadratic function can be rewritten into the form $$Q(\vec{x}) = ...
0
votes
2answers
49 views

Describe the transformation of $F(x)=x^2$ represented by $G(x)=(x+4)^2$

I am completely confused in my math class and I was wondering if someone could help me on this question by explaining it to me step by step? The question looks like this. Describe the ...
0
votes
1answer
26 views

Solving the following equation by factorisation

The given equation is, $\frac{m}{n}x^2+\frac{n}{m}=1-2x$ What I've tried, Multiplying the equation by $n$, we get $mx^2+\frac{n^2}{m}x=n-2nx$ Now what? I am completely confused about what to do. ...
0
votes
2answers
69 views

What will be the sign of the constant term ‘c’?

We have a quadratic equation $ax^2$+ $bx$ + c =0 . It has no real roots. i.e $b^2$- $4ac$ < 0 .What is the sign of c ? My tries- I have supposed the $4ac$ will be greater than $b^2$ , because ...
-3
votes
6answers
85 views

How to find $c$ and $d$ from the equation $(c+id)^2=1$? [closed]

I need to solve this complex equation: $$ (c + id)^2 = 1 $$ where $i^2=-1$. What am I supposed to calculate here? Just $c$ and $d$?
0
votes
0answers
43 views

How to solve this system of equations with five unknowns?

I want to know if there are exact solutions to the following system of five equations with five unknowns $A,B,D,F,R$ , with a step by step solution $AD=32$ $AF+BD=16$ $AR+BR+D=-32$ $BR+F+A=-12$ ...
2
votes
2answers
43 views

Calculating the roots of a quadratic with complex coefficients

$x^2$-(5i+14)x+2(5i+12)=0 I got : $\frac{(5i+14)+(75+100i)^{1/2}}{2}$ and $\frac{(5i+14)-(75+100i)^{1/2}}{2}$ Wolfram gives : 2 and 12+5i How do I reduce my solutios?
0
votes
3answers
28 views

Algebraic Solution to $z^2 = az + b^2$

In a book about the history of invisible numbers, the author writes: $\frac 1 2 a + \sqrt {(\frac a 2)^2 + b^2}$ is the solution to $z^2 = az + b^2 $ Where is this coming from? I could not find a ...
2
votes
1answer
201 views

General method for determining if $Ax^2 + Bx + C$ is square

Is there a general method for solving Diophantine equations in the form $Ax^2 + Bx + C = k^2$, preferably turning them into Pell's equations, when possible? For example, $2x^2 + x + 1 = k^2$ or $5x^2 ...
3
votes
3answers
168 views

How can I find the quadratic formula using calculus?

Context: A few years ago, when I was in senior year, I participated in the Student for a Day at my local CEGEP (in Quebec, there are just 5 years of middle and high school.A CEGEP is a school you go ...
1
vote
2answers
34 views

A polynomial problem related to lx^2 + nx + n

If the roots of $lx^2 + nx + n = 0$ are in the ratio $p:q$, find the value of $\sqrt{\frac{p}{q}}$ + $\sqrt{\frac{q}{p}}$ + $\sqrt{\frac{n}{l}}$. How to go about this problem?
3
votes
1answer
152 views

If $|ax^2+bx+c|\le 1\ \forall |x|\le 1$, then what is the maximum possible value of $\frac 83a^2+2b^2$? [closed]

Let $f(x) = ax^2 + bx + c$ ; $a,b,c\in\mathbb R$ It is given that $|f(x)| \le 1$ $\forall |x| \le 1$ Q1) The possible value of $|a+c|$, if $\displaystyle \frac{8}{3} a^2 + 2b^2$ is maximum, is ...
1
vote
3answers
73 views

Finding the matrix of a quadratic form

I want to find the matrix of quadratic form $Q= \sum^p_{i=1} (y_i - \bar y)^2$. Please help me finding it. For example I have found the quadratic form matrix for $Q= p\bar y^2$ as follows: ...
1
vote
1answer
37 views

Find an equation of the quadratic function with zeros at $(0, 0)$ and $(6, 0)$ with $f(5) = -15$

The Question is: write the equation of the quadratic function with zeros at $(0,0)$ and $(6,0)$ with $f(5) = -15$. So, I know how to get the equation from the zeros, but I am confused with what I am ...
0
votes
3answers
42 views

$x^2+y^2+2axy=0 \Rightarrow x=0$ and $y=0$

Show that for all real numbers $x$ and $y$, for all $-1<a<1$ $$x^2+y^2+2axy=0 \Rightarrow x=0 \text{and} y=0$$ I see that $x^2+y^2+2axy=(ax+y)^2+x^2-(ax)^2$. I'm stuck here.
0
votes
1answer
31 views

How to solve the following trigonometric quadratic equation for x

How to solve the following trigonometric quadratic equation for x $3 \cos x + r \cos^{2} x - 2 \sin x -r \sin^{2} x = 0$ where r is a constant Even though this trigonometric quadratic equation has ...
0
votes
1answer
37 views

Sum of equation's roots

Find the sum of all real roots of all polynomials in the form of $y=x^2+px+122$, where $p$ can be all integers in the range $[-35,17]$ I could only do it via excel) Which is not mathematical proof.. ...
1
vote
2answers
24 views

Possible values for this specific line of variables.

I have this line of numbers: xy + z = xz + y = yz + x I need to find out all the possible values of x, y and z in this equation. Thank you!:) My usual problem ...
3
votes
0answers
86 views

How can I find values for which a given expression gives a perfect square?

There have been several posts on this topic on math.se, such as this one with the same title. However all the posts I found contained coefficients to $x^2$, that were perfect squares. I am looking for ...
0
votes
3answers
56 views

how to solve this: $z^2-(1-3i)z-2i-2=0$

I've tried two ways, but get stuck. I've tried to simplify, but didnt know what to do next, and i've tried to solve it like Quadratic equation, but got stuck too. tnx.. one way got me this: z/2 * ...
0
votes
1answer
31 views

If $4a^2+9b^2-c^2+12ab=0$,the family of straight lines ax+by+c=0 is concurrent at which point?

If $4a^2+9b^2-c^2+12ab=0$,the family of straight lines $ax+by+c=0$ is concurrent at which point? How to solve such problems.Hints please!
2
votes
2answers
33 views

Proving homogenous quadratic inequalities

Okay, I'm having trouble proving this: $$5x^2-4xy+6y^2 \ge 0, \text{ where } x,y \in \mathbb{R}$$ I have tried a few values of $x$ and $y$ and I find that is true. EX: $x=y=0$ which make the equation ...
1
vote
3answers
45 views

Simultaneous equations - How does the following make sense

The following example is derived from the video https://www.youtube.com/watch?v=xee7Qqqd2eU , At the 5 minute mark the two corresponding lines of work are as follows:- $$2x^2-x-10 = 0$$ equal to ...
6
votes
4answers
533 views

How to prove that the following system of equations has only one solution?

$ \begin{cases} (x - 1)^2 + (y + 1)^2 = 25 \\ (x + 5)^2 + (y + 9)^2 = 25 \\ y = -\frac{3}{4}x - \frac{13}{2} \end{cases} $ I have to solve this system of equations. After substituting $y = ...
1
vote
1answer
14 views

Method for transforming one curve around another?

I'm working with a complex problem involving waveforms. Essentially I want to bend a given waveform around a circle. At it's most basic, I want to take one curve on a linear graph and map it onto a ...
-4
votes
3answers
97 views

Solving $\sqrt{25 - 10x + x^2} = x-5$ [closed]

I need help solving this math question. $$\sqrt{25-10x+x^2}=x-5$$ I got $x=5$, but apparently it is wrong. Please provide an explanation if possible. Thanks.
0
votes
1answer
456 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
3answers
53 views

Why does a root only have a positive output? [duplicate]

Let's say I am solving an equation, and end up with this: x^2 = 16 The solutions will be x=4 or x=-4 That makes sense. But when I have this: x = √16 The only solution is x=4 ...
1
vote
2answers
112 views

Why can I not use the quadratic formula here?

$x^2-14x+49$ I tried applying the quadratic formula here, but I end up with $14 ± 0$ which makes no sense. I am supposed to factor the above mentioned expression and I thought the quadratic formula ...
4
votes
0answers
60 views

Symmetric proof for the probability of real roots of a quadratic with exponentially distributed parameters

What is the probability that the polynomial has real roots? asked for the probability that the quadratic polynomial $ax^2+bx+c$ has real roots if the parameters $a,b,c$ are exponentially distributed ...
5
votes
1answer
97 views

Number Theory: $x^2\equiv 1\pmod{140}$

I have this problem assigned for homework and I'm confused as to how to solve an $x^2$ congruence. Here is the problem: $x^2\equiv 1\pmod{140}$ My only thought was to do something along the lines ...
0
votes
1answer
22 views

find the solution set of the following equation with absolute value

WA online me results $x = -5$, failed to reach it $$\left ( \left | x \right |+2 \right )\left ( \left| x-2 \right| -3\right)=x^{2}+3$$
-1
votes
1answer
86 views
1
vote
6answers
67 views

How do you factorize quadratics when the coefficient of $x^2 \gt 1$?

So I've figured out how to factor quadratics with just $x^2$, but now I'm kind of stuck again at this problem: $2x^2-x-3$ Can anyone help me?
0
votes
0answers
28 views

quadratic inequality with parameter solving

Given $(E_m): (m-1)x^2 + 2(m+1)x + m - 5$ where m is a real parameter, and designate by $x_1$ and $x_2$ the roots of this equation when they exist. on the axis $x'ox$ given the points $M_1$ and $M_2$ ...
1
vote
1answer
194 views

Show that a quadratic function is always positive for all real values of $x$

How can I show that $x^2 +x +1$ is aways positive for all values of $x$? Do I use discriminant or completing the square?
0
votes
1answer
54 views

Sum and Product Of Roots

If α and β are the roots of the equation x2 + px -q =0 and γ and δ are roots of x2 +px+r =0 then the value of (α-γ)(α-δ) is:- (Choose the correct option) p+r p-r q-r q+r
0
votes
1answer
36 views

Quadratic Equation by Factoring

My son has this problem on his homework and I can't figure out the process to solve this equation. The instructions say "Solve each equation by factoring". Can someone help and please show steps? ...
1
vote
3answers
37 views

Common roots in a quadratic and a cubic polynomial

Let $f(x)=x^3-3x+b$ and $g(x)=x^2+bx-3$ where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x)=0$ and $g(x)=0$ have a common root? I just need a hint ...
0
votes
1answer
18 views

Quadratic equation problem

Quadratic eq which takes the value (y) = 41 at x= -2 , the value (y) = 20 at x =5 when x = 2 -> a4-b2+c=y ... how to find a,b,c,y here is original question: ...
25
votes
6answers
1k views

Definition of “simplify”

In mathematics the word "simplify" is used a lot. In a lot of cases it is obvious what actually makes an expression simpler, but not always. Is there a measurable definition of simplicity or is it ...