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Questions tagged [quadratics]

Questions about 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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votes
3answers
40 views

Question regarding the roots of $x^2 + nx + m$, for $n,m\in\mathbb{Z}$.

The equation $x^2+nx+m=0$, $n,m\in\mathbb Z$, cannot have A. Integral roots B. Non-integral roots C. Irrational roots D. Complex roots Please explain also....
3
votes
1answer
46 views

Quadratics: Intuitive relation between discriminant and derivative at roots

While working with quadratics that have real roots, I realized an interesting fact: The slope of a quadratic at its roots is equal to $\pm \sqrt{D}$ where $D=b^2-4ac$ Proof: $$f(x) = ax^2 + bx +c$$...
-2
votes
1answer
21 views

Quadratic equation - finding $x$ given $y$ value

I've been having trouble with this quadratic equation where $6639.55 = -0.06493x^2 + 22.35175x + 5065.25$. My question is, how do I find out $x$? Thank you in advance for any advice!
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votes
0answers
30 views

What is 15 and 290 theorem ,can anyone answer me ,please? [on hold]

A theorem in Quadratic forms which related to the number theory.
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votes
1answer
31 views

How do I solve this trajectory of projectile equation? [closed]

$$y\ =\ x\tan\theta\ -\ \frac{1}{2}\cdot g\cdot\frac{x^2}{v_0^2.\cos^2\theta}$$ Where $$g\ =\ 3;$$ & $$y\ =\ 3\ @ \ x\ =\ 3,$$ $$y\ =\ 0\ @\ x\ =\ 6,$$ $$y\ =\ 0\ @\ x\ =\ 0,$$ Solve For: $$\...
-1
votes
1answer
26 views

Condition for two cubic equations to have a pair of common positive repeated roots

I was asked to find the condition that two cubic equations $a_1x^3 + b_1x^2 + c_1x + d_1 = 0$ and $a_2x^3 + b_2x^2 + c_2x + d_2 = 0$ have a pair of common positive repeated roots. The answer given is ...
0
votes
4answers
75 views

Age problem-PLEASE HELP [closed]

Hannah. added one to her age, then multiplied the result by $4$. She squared that product, then increased the result by $3$, then doubled it. The final result was $1214$ more than $60$ times her age. ...
3
votes
2answers
82 views

Advice on solving these simultaneous (quadratic/cubic) equations?

I have the following simultaneous equations: $$a x^2 + (b+2ay)x - c_1 = 0$$ $$ay^2 + (b+2ax)y - c_2 = 0$$ Where I'd like to solve for $x$ and $y$. Obviously $a,b,c_1,c_2$ are known constants. They ...
5
votes
3answers
389 views

Easier way to find amount of solutions between a line and quadratic?

Is there a better way of finding the number of solutions of the system: $y=(x-7)(3x+4)$ and $x=3y-1$ that doesn't involve calculus? I know these are $2$ but that's due to substitution of one equation ...
2
votes
3answers
38 views

Find the exhaustive values of “a”, given the condition

If the equation $2^{2x} + a*2^{x+1} + a + 1=0$ has roots of opposite sign then the exhaustive values of a are? I tried taking $2^x = t$. But then didn't know what to do. The equation became, $t^2 + ...
0
votes
2answers
41 views

How to solve the following system of equations?

$\left\{ \begin{aligned} xy + 2x + 2y &= -8\\ yz + 2y + 2z &= 24\\ xz + 2x + 2z &= -11 \end{aligned} \right.$ I need to solve it over the set of real numbers.
1
vote
1answer
75 views

Is it possible to isolate the variable X in this equation? [closed]

$P = \dfrac{1}{\sqrt{x^2+xac+a^2}} + \dfrac{1}{\sqrt{x^2+xbc+b^2}}$
0
votes
0answers
29 views

Quadratic Simultaneous Equations with Four Variables

I have the following equations, where $a$ to $j$ are real constants, and $w, x, y$ and $z$ are values to be solved for: $$awx - bwy - cxy + dy^2 = 0$$ $$gyz - bwy - fwz + ew^2= 0$$ $$gyz - ixz - cxy +...
2
votes
0answers
28 views

Is it possible to find a solution to the following $(x(b+x) + c - f) \mod (2x+2) = 0$

Given the equation $$ (x(b+x) + c - f) \mod (2x+2) = 0 $$ Is it possible and if so what is the quickest way to find appropriate value of $x$? Where $x > 0$ The above equation is derived from the ...
-1
votes
3answers
46 views

Minimum possible number of positive root of the quadratic equation $x^2-(1+\lambda)x+\lambda-2=0,\lambda\in R$ is

Minimum possible number of positive root of the quadratic equation $x^2-(1+\lambda)x+\lambda-2=0,\lambda\in R$ is $(a)2$ $(b)1$ $(c)0$ $(d)3$ $x^2-(1+\lambda)x+\lambda-2=0$ I changed this equation to ...
6
votes
4answers
59 views

If $a,b\in R$ are distinct numbers satisfying $|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$,then find the minimum value of $|a-b|$

If $a,b\in R$ are distinct numbers satisfying $|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$,then find the minimum value of $|a-b|$ I squared both sides of $|a-1|+|b-1|=|a|+|b|$ and $|a|+|b|=|a+1|+|b+1|$ and ...
1
vote
1answer
37 views

Confusion with binomial factorization

Say you have the term: $-3x^2+12$ and you want to factor it out: $$-3(x^2 - 4)$$ $$-3((x+2)(x-2))$$ However you can also write the above as: $$-3(x+2) (x-2)$$ But in many other cases, e.g if you ...
1
vote
2answers
30 views

Is the set of all quadratic, real-valued functions a vector space?

I know there are already some similar questions but I still do not understand how to prove that the set of all quadratic functions defined on the real line is a vector space. The question I got was: "...
2
votes
2answers
41 views

If the equations $ax^3+(a+b)x^2+(b+c)x+c=0$ and $2x^3+x^2+2x-5=0$ have a common root,then $a+b+c$ can be equal to

If the equations $ax^3+(a+b)x^2+(b+c)x+c=0$ and $2x^3+x^2+2x-5=0$ have a common root, then $a+b+c$ can be equal to(where $a,b,c\in R,a\ne0$) $(1)\;5a$ $(2)\;3b$ $(3)\;2c$ $(4)\;0$ As $x=1$ is the ...
19
votes
5answers
2k views

Mistake in solving an equation involving a square root

I want to solve $2x = \sqrt{x+3}$, which I have tried as below: $$\begin{equation} 4x^2 - x -3 = 0 \\ x^2 - \frac14 x - \frac34 = 0 \\ x^2 - \frac14x = \frac34 \\ \left(x - \frac12 \right)^2 = 1 \\ x ...
0
votes
1answer
13 views

Explanation of gradient descent on convex quadratic

Can someone explain the following: $$f(x) = \frac{1}{2}w^TAw - b^Tw$$ Assume AA is symmetric and invertible, then the optimal solution $w^{\star}$ occurs at $$w^{\star} = A^{-1}b$$ and $$\nabla f(w)...
0
votes
3answers
56 views

Proof that equation does not have real roots

For polynom $f(x)=ax^2+bx+c$ equation $f(x)=x$ has no real solutions. Prove that equation $ f(f(x))=x$ also does not have does not have real solutions Can someone explain solution to me? Why? ...
0
votes
3answers
45 views

If $a$ and $b$ are integers s.t. $2x^2 -ax + 2 > 0$ and $x^2 -b x + 8 \geq 0$ for all real numbers $x$, then the largest possible value of $2a−6b$ is

If a and b are integers such that $2x^2-ax+2>0$ and $x^2-bx+8≥0$ for all real numbers $x$, then the largest possible value of $2a−6b$ is Answer: 36. My attempt: Multiply the first inequality by 2 ...
1
vote
1answer
37 views

Silly Question on Definition of real solutions and real roots

For $a\neq0$, the equation $ax^2+b|x|+c=0$ has k real solutions and p real roots. Here, my doubt is what does the solution means and how it is different from real roots? Thanks for clearing my ...
0
votes
1answer
18 views

Solving a quadratic equation for ellipsoid

Suppose we have the equation \[ \langle a - Ax, B(a - Ax) \rangle \leq t \] where $a,x \in \mathbb{R}^n$ and $A,B \in \mathbb{R}^{n\times n}$ are invertible. Is there a way I could solve the above to ...
3
votes
5answers
49 views

Find all the solutions of $z^2-(1+3i)z-8-i=0$

I am stuck on a problem and I was hoping someone could tell me what I am doing wrong. I want to find all the roots of: $$z^2-(1+3i)z-8-i=0$$ There are two ways I tried to approach this. ...
0
votes
2answers
33 views

Maximum value of a function with 2 variables

Can someone help me finding maximum value of a ratio in quadratic function in 2 variables using proper mathematical methods.? Question is as below. If x and y are real numbers such that $x^2 -10x+...
0
votes
1answer
13 views

The greatest area for a rectangle on a track field.

An athletic field with a perimeter of 0.25 miles consists of a rectangle with a semicircle at each end, as shown below. Find the dimensions that yield the greatest possible area for the rectangular ...
0
votes
1answer
36 views

Quadratic Functions - vertex of graph proof

Can you explain how $x$ becomes $x + b/2a$ and $c$ becomes $4ac - b^2/4a$ all of sudden? Can you please explain at a Pre-Calculus level, thank you very much.
0
votes
1answer
33 views

Discriminant when graph lies above or below the x axis.

Suppose a quadratic equation has been given where the a value (ax^2 + bx + c) is a positive and it has been said that the graph of the equation lies above the x-axis- what is the discriminant? For ...
0
votes
1answer
19 views

How to know if a graph is exponential just by looking at the data values

If I were given the points (1,3) (2,5) (3,7) and assumed this pattern continued forever, I know that it is linear as there is a constant the y value for an increase of one for the x value. If I were ...
0
votes
0answers
18 views

Intervals of a Multivariable Function

If the gradient at some point of a multivariable function equals $\vec{0}$, and the Hessian is positive or negative semidefinite, is there a notion, as in single variable calculus, of resolving the ...
3
votes
1answer
132 views

Finding the minimum value.

I'm struck on this question, I tried hard but couldn't solve it. Question: if a quadratic equation in $x$: $$ax^2 - bx + 5 = 0$$ does not have two distinct real roots, then find the minimum value of $...
0
votes
1answer
19 views

If the parabola is translated from its initial position to a new position by moving its vertex along the line $y=x+4,$

The parabola $y=4-x^2$ has vertex $P.$It intersects $x-$axis at $A$ and $B.$ If the parabola is translated from its initial position to a new position by moving its vertex along the line $y=x+4,$ so ...
0
votes
3answers
47 views

Find all $a, b \in \mathbb R$, ($b\ne0)$, such that the roots of $x^2+ax+a=b$ and $x^2+ax+a=-b$ are 4 consecutive numbers

Find all $a, b \in \Bbb R$, ($b\ne0)$, such that the roots of $$x^2+ax+a=b$$ $$x^2+ax+a=-b$$ are 4 consecutive numbers. We have: $$x^2+ax+a-b=0$$ $$x^2+ax+a+b=0$$ $x_1, x_2$ - roots of first equation;...
0
votes
4answers
38 views

Having issues understanding fraction division when applying quadratic formula

I'm trying to apply the quadratic formula, and having trouble understanding how: $$\frac{-3 ± 3\sqrt{41} }{-18}$$ evaluates to $$\frac{-1 ±\sqrt{41}}{-6}$$ and not $$\frac{1}{6}±\frac{\sqrt{41}}{...
0
votes
4answers
41 views

When I complete the square on $3x^2 - 12x + 14$ I get an imaginary number, where have I gone wrong?

I have a question in my excersise book: By completing the square show that the expression $3x^2 - 12x + 14$ is positive for all $x$ My approach was to complete the square and rearrange to make $x$ ...
1
vote
2answers
73 views

How do you turn an irrational, non-transcendental number, like 1.618… back to its form of (a + sqrt(b))/c.

Looking at irrational numbers, I had an idea, as to computing square roots. Take the golden ratio. Numerically, it's 1.618.... but I can also write it like this: $\frac{1+ \sqrt{5}}{2}$ I want to ...
1
vote
2answers
35 views

How to factorize $zz^*-4z-4z^*+12=0$ (where $z^*$ is the complex conjugate of $z$)

I'm trying to factorize this: $$zz^*-4z-4z^*+12=0$$ to get this: $$|z-4|^2 - 4 = 0$$ where $z=x+yi$ is a complex number and $z^*=x-yi$ is the conjugate complex number of $z$. I'm trying to factorise ...
1
vote
1answer
42 views

Piece-wise quadratic function [closed]

How can I find a minimum of a piece-wise quadratic function? (minorant of a set of quadratic functions) An example of this will be appreciated.
1
vote
5answers
62 views

Information lost in solving system of quadratic equations

I have a system of two quadratic equations $$ \left\{ \begin{array}{c} 2x^2+x-1=0 \\ 2x^2+5x+2=0 \end{array} \right. $$ I tried to solve it the following way: $$ 2x^2=-5x-2$$ substituting in the ...
1
vote
1answer
28 views

The Shading of Double Petaled Flowers

Here is the shape that I am trying to shade in. I have the outline. Here are the equations that I used: I was wondering how I could manipulate these domains and ranges or maybe the equations. I was ...
0
votes
3answers
37 views

Find the equation of a line intersecting a parabola

Okay here's the question: Consider the parabola P of equation $y=x^2$, and the line $L$ of equation $y=x+6$. Let $P(x_p,y_p)$ be a point on the arc of the parabola P below L. Let A and B be the ...
-2
votes
1answer
9 views

Quadratic Yield Response Function [closed]

How do I find the quadratic yield response function in the form of "Y = b0 + b1*X + b2*X^2" for a set of data in excel?
2
votes
0answers
48 views

Show that this system has at least one unbounded solution as $t \to \infty$

Assume the system $$x'(t)=\begin{pmatrix} \frac12-\cos t & 2 \\ 1 & \frac32+\sin t \end{pmatrix}x(t)=A(t)\cdot x(t)$$ with minimum period: $T=2\pi$. Let $\mu_1,\mu_2$ be its characteristic ...
1
vote
2answers
45 views

Show that $|\frac{1}{2n}-\frac{1}{2m}| < \epsilon$ holds for all $m, n > \frac{1}{\epsilon}.$

In Example 1.5-9 of the book Functional Analysis by Kreyszig it claims that $|\frac{1}{2n}-\frac{1}{2m}| < \epsilon$ holds for all $m, n > \frac{1}{\epsilon}.$ My calculations don't lead to ...
-1
votes
6answers
33 views

How would one factorise $m^2 + (2AB)m + B^2 =0$

How would one factorise $m^2 + (2AB)m + B^2 =0$, to go onto solve a second order differential equation
0
votes
1answer
41 views

How to graphically depict the possible solutions of a quadratic equation

I have the following quadratic equation : $$am^2 + bm + (c_1^2 +c_2^2) =0,$$ where the solution is given by $$m = \frac{-b\pm\sqrt{b^2-4a(c_1^2+c_2^2)}}{2a}.$$ Here, $\Delta>0$. Thus I have ...
0
votes
1answer
28 views

Is there a formal name for an equation with multiple solutions?

I saw that there is a related question for an equation with no solution, but I was curious about an equation with more than one solution.
0
votes
1answer
40 views

Root of an quadratic equation

I have the following quadratic equation : $m^2 + m(p-1/l) - (\Omega_x^2 + \Omega_y^2)=0$ I would like to get the solution in terms of $\Omega_x, \Omega_y$ with some approximations i.e. neglecting $(...