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)

-3
votes
6answers
84 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
41 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
42 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
26 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
195 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 ...
2
votes
3answers
134 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
31 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
139 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
66 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
36 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
41 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
30 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
83 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
54 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
26 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
44 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
500 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
13 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
95 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
125 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
50 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
111 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
51 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
0answers
91 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
18 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
77 views
1
vote
6answers
65 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
23 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
87 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
52 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
0answers
27 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
33 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 ...
0
votes
3answers
28 views

Using the quadratic function to make a prediction

I've been given a simple problem: use the quadratic function to predict the number of subscribers expected in 2010. According to the table, $$\begin{array}{|c|c|} \hline \text{Year} & ...
1
vote
3answers
71 views

Finding condition for integral roots of a quadratic equation.

I need to find the values of k(possible) for which the quadratic equation $$x^2+2kx+k =0$$ will have integral roots. So I assumed roots to be $a,b$ Then I got the condition $a+b=-2k$and $a\cdot b=k$; ...
0
votes
2answers
42 views

Logarithmic equation 3^(x+1)+9^(x-1)-61236=0 produces unclear solution

I have a basic logarithmic equation 3^(x+1) +9^(x-1) -61236 =0 which I simplified like ...
-2
votes
2answers
64 views

SAT Math Question: Long division of polynomials [closed]

$$P(x)=3(x^2+10x+5)-5(x-k)$$ In the polynomial $P(x), k$ is a constant. If $P(x)$ is divisible by $x$, what is the value of $k$?
0
votes
4answers
69 views

Finding the real value of x

I replaced the numbers with variables so i let $y=2$ and $z=12$ and i got the following after simplifying the left side: $$2^{144}-\frac{1}{2^{144}}=8^x-8^{-x}$$ but i'm not seeing how this can ...
0
votes
0answers
12 views

Is it possible to rewrite the following system in a matrix form?

$x_i\mathbf{(x^T-A_{i\cdot})z_i}=\mathbf{A_{i\cdot}z_ix^Tz_i}$ where $\mathbf{x}$ is a $p$-by-$1$ vector, $x_i$ its $i$-th entry, $\mathbf{z_i}=[1,1,\dots,0,\dots,1]^T$, a $p$-by-$1$ vector of which ...
0
votes
0answers
11 views

Representation of integers by sum of squares with linear constraints of a special form

I would like to know what integers $d$ can be written as a sum $d=\sum_{i=1}^N\sum_{j=1}^M a_{ij}^2 $ with $a_{ij} \in \mathbb{Z}$ and where the row and column sums of $a_{ij}$ are fixed $\sum_{i=1}^N ...
0
votes
1answer
42 views

Factoring quadratic equations

During the video on the link (at 20 seconds) the narrator says that $(x^2+3)$ cannot be factored, however I believe that it can be factored to $(x-1)(x-3)$ https://www.youtube.com/watch?v=4IeZkmO0STg ...
2
votes
1answer
47 views

How to solve this integral which has a root over the quadratic polynomial divided by another quadratic polynomial?

Can anyone help me solve this integral? I have the value of $b$ (real and positive, most probably lower than $5$) and using the value of integral, I have to calculate $a$. $$ ...
0
votes
1answer
53 views

Reducing equations to quadratic form

I have a chapter in my school course book on quadratic equations, in which we are learning how to solve nonquadratic-equations , by reducing them to quadratic form, the book describes 5 types of ...
0
votes
1answer
56 views

What is the geometric interpretation of the leading coefficient in a quadratic equation?

What is the geometric interpretation of the leading coefficient in a quadratic equation? It is clear we do not call it a slope or elasticity or derivative of the quadratic equation.
2
votes
1answer
14 views

Interval in which roots lie given inequality between coefficients

Given that in a quadratic equation $ax^2+ bx + c=0$, $(4a+c)^2<4b^2$, Find the interval in which roots lie. I subtracted $16ac$ from both sides, to get $\Delta>(4a-c)^2$, which is always ...
0
votes
1answer
59 views

Solving a system of equations with Cobb-Douglas production function

I have two equations and two unknowns in the following: $$p \alpha x_{1}^{\alpha-1}x_{2}^{\beta}-w_{1}=0,$$ $$p \beta x_{1}^{\alpha}x_{2}^{\beta-1}-w_{2}=0.$$ After solving I am supposed to get my ...