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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2
votes
4answers
105 views

Finding the values of $5a+b$ if $ax^2+bx+10$ does not have $2$ distinct roots

If $ax^2+bx+10=0$ does not have two distinct real roots where( $a$ and $b$ are real) then the possible values of $5a+b$ from the given 4 options(as obviously there can be infinite possible values..but ...
0
votes
2answers
35 views

Finding common interest rate

My question is; How can I approach a common interest rate? (See example below for an explanation) Consider the example in the table below ...
-4
votes
2answers
55 views

How to prove that the roots of $(m-2)x^2-(3m-2)x+2m=0$ are real

I need help proving that the roots of the equation: $(m-2)x^2-(3m-2)x+2m=0$ are real. Could you also give me a step by step runthrough of how to do this equation? (I have a few others like this one to ...
0
votes
0answers
11 views

Is this the correct way to interpret results from the principal minor test?

Am I correct in thinking that, given all the principal minors $A_{i}$ of the matrix associated with a quadratic form $Q(x,y)$, the Principal Minor Test states that $Q(x,y)$ is Positive Definite if ...
0
votes
0answers
19 views

How Do I Obtain X from this equation?

I tried different ways, but I don´t succeed: $y=\frac{4x-1}{8x^4}$ In one of them I reach this point: $4x(2x^3-1)=-\frac{1}{y}$
2
votes
1answer
49 views

Prove a binary quadratic equation has specific number of solutions

How do I show that the binary quadratic equation $f(x, y) = x^2 + xy + y^2 = 1$ has exactly $6$ solutions? The discriminant is $-3$, so I cannot use Pell's Equation ($x^2 - dy^2 = p$, where $d>0$ ...
1
vote
4answers
45 views

Basic quadratic calculus

Given $$ AX^2+2X-1=0 \,\ \text{where} \,\ A>0$$ What value of A would make the absolute value of both roots bigger than $1$? I found the roots using quadratic formula and also found that A has to ...
0
votes
0answers
59 views

Reference book for quadratic equations with hard exercises?

I'm teaching advanced level mathematics. And I need good question banks on quadratic equations . More likely good question bank on advanced level maths.
1
vote
0answers
31 views

Problem on Bivariate normal distribution

Let $X_1$ and $X_2$ have a bivariate normal distribution with parameters $\mu_1 = \mu_2 = 0$ and $\sigma_1 = \sigma_2 = 1$ and $\rho = 1/2$ Find the probability that all the roots of $X_1x^2+ 2X_2x + ...
1
vote
0answers
33 views

How to minimise objective function with H1 constraint?

I have a cost function that I try to minimise: $$ \Pi \sim ||y - Ax||^2 + ||\nabla x || ^2 $$ The gradient is there to constraint that my solution at minimum $x_{min}$ is continuous in 1st ...
0
votes
2answers
33 views

Correct notation for inequality equation

I'm currently taking a Polynomial functions -course and learning about quadratic inequality equations. Let's consider this equation: $-x^{2}+7x<10$ The correct answer to this according to our ...
0
votes
2answers
83 views

Need help with a certain integral

Sorry, this is my first post ever and formatting is bad. I appreciate all assistance Given the equation y= $ 3.8x^2-4.4x+1444 $ Find the definite arc length integral between 0 and 1200. I am not ...
0
votes
0answers
44 views

Find root of $x^2 + bx + c$ with an approximation for $b$

I have a monic quadratic equation $x^2 + bx + c$, where $b, c \in \mathbb{Z}$ for which I need to compute the roots. $x$ is also an integer. However, I don't have the exact value of $b$, only an ...
1
vote
2answers
24 views

Showing solutions for this recurrence relation

I am having trouble starting this question. Given that the sequence ($x_n$) satisfies the recurrence relation: $x_n$$_+$$_1$ = a$x_n$ + b$x_n$$_−$$_1$, where $n= 1, 2, ...$ and $a$ and $b$ are given ...
0
votes
0answers
22 views

Solving non-commutative “quadratic” equation with inhomogenously typed coefficients

Is there a general method to solve for $z\in\mathbb{R}^d$ in the non-commutative $z^\intercal\alpha z + \beta z + \gamma = 0$ where $\alpha\in\mathcal{M}_d(\mathbb{R})$ (real $d\times d$-matrix), ...
0
votes
1answer
65 views

How to solve a system of quadratic equation as follows?

I have a system of quadratic equations: \begin{align*} v_2x_1^2 + v_2x_1 - v_1x_2^2 - v_1x_2 & = 0\\ v_3x_1^2 + v_3x_1 - v_1x_3^2 - v_1x_3 & = 0\\ v_3x_2^2 + v_3x_2 - v_2x_3^2 - v_2x_3 & = ...
1
vote
0answers
42 views

Does an equation of this type have complex solutions?

How can I show, that every quadratic equation of the type $z^2+az+b=0$ with complex coefficients $a,b$ has a solution in $\mathbb{C}$? And how can I get these solutions? At first, I tried applying ...
0
votes
2answers
48 views

solution of quadratic equation n unknown

My question is as follows: Let the equation $V^{\top}MV=F$. Such as $V^{\top}=(x_1,x_2,...,x_n)$ a line vector of n unknown coefficients, M a known diagonal matrix (of size n) and F a real number ...
0
votes
1answer
16 views

Conditions present after solving for x

The question asked me to solve $\dfrac {x^2+2x-8}{x^2-x-2}=3$ The answer is $x=0.5$, which I worked out, but in the answers it says that I must state the condition that $x≠2$ to get full marks. Why ...
5
votes
3answers
130 views

Solving $10x^4-13x^2+4=0$

I just came across a question in my paper that asks me to solve for $x$ in $10x^4-13x^2+4=0$ I've only learned how to factorize quadratics and the quadratic formula, but I'm not sure how to factorize ...
0
votes
3answers
16 views

Factorising quadratic equation with an unknown given 1 root

In a algebra quadratic question, it says: The equation $3x^2+4x-k=0$ has two distinct real roots. If 2 is a root of this equation, find the value of $k$ and the second root. The first line of ...
0
votes
0answers
23 views

Principle axes rotation.

I am trying to rotate a quadratic function to its principle axes. I have transformed the function to a symmetric 3x3 matrix I am struggling to know what to do next I thought to rotate the matrix but ...
1
vote
1answer
28 views

Rewriting an algebraic equation with square roots

In as part of solving a question, the equation $a-3\sqrt a-4=0$ is written into $a^2-3a-4=0$ How is this done? Do you square everything in the equation? But in this case why are only the $a$ ...
0
votes
0answers
13 views

Keeping a parabola's roots after vertical shift

Suppose f(x) = -x(x - a) + b, where a > 0 and b >= 0. When ...
0
votes
4answers
90 views

How to Solve $\frac{(x^2+1)}{x} + \frac{x}{(x^2+1)} = 2.9$?

My cousin has this math homework problem she is stuck on and I have also failed at it. Maybe you guys can help? $$\frac{x^{2}+1}{x} + \frac{x}{x^{2}+1} = 2.9$$ The answer is $0.5$ or $2$, but we're ...
0
votes
0answers
18 views

polynomial roots and density

Let $f(x)=x^2+2x$ be a quadratic polynomial and $\mathcal{R}_f$ be its range. Denote $\mathcal{A}_{i+1}$ be the set of real numbers that lie in $\mathcal{R}_f$ and come from finding roots of ...
0
votes
0answers
48 views

Econometrics: quadratic specification, turning point and time-series

Given the following quadratic specification : ln(yt) = c + beta1*ln(xt) + beta2*ln(xt)^2 where t: represents time Ln: natural logarithmic c: constant yt: dependant variable at time t xt: ...
0
votes
0answers
13 views

Manipulation of this equation into a Gaussian form

I have been going through a paper (http://www.jting.net/pubs/2007/ting-ICRA2007.pdf) and trying to work out the maths. Ultimately, I came to the following expression $$ \bigg[\frac{1}{2 ...
-1
votes
2answers
45 views

Is is possible for a quadratic equation with only one irrational root to have integral coefficients?

Given a quadratic equation with one and only one root (for example $6-\sqrt{2}$ ). Does there exist integers $a,b$ and $c$ where $ax^2 + bx + c = 0$ for that root?
0
votes
1answer
51 views

Quadratic equation involving right-angled triangle

I have right angled triangle I've been attempting to prove a quadratic equation with for a while. It has a hypotenuse of $2x + 1 cm$, a base of $x + 5 cm$, and height of $x - 2 cm$. I calculated its ...
0
votes
1answer
32 views

How to solve this equation for y? $y^2+2yx-3x^2=0$

How to solve this equation for y? $$y^2+2yx-3x^2=0$$ I know how to factor the unknown y with the coefficients of the expression and it would be $(y+3)(y-1)$ but I don't remember how to work out x. ...
0
votes
1answer
46 views

Quadratic function.

Let $\quad f(x)=ax^2+bx+c \quad $ be a quadratic function, $x \in \left (0,d \right )$ a domain of $f(x)$. If I need $f(x)$ to be positive, that would be achieved for $ x \in \left (-\infty,x_{-} ...
0
votes
2answers
52 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
39 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
19 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
48 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
61 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
32 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
29 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
71 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
86 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
44 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
208 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 ...