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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How surfaces intersect in projective spaces

Consider this parametrization $$\phi:\mathbb{P}^1\longrightarrow\mathbb{P}^3$$ $$(t_0:t_1)\longmapsto (t_0^3: t_0^2t_1:t_0t_1^2:t_1^3)$$ Let $\mathcal{C}$ be the image of $\phi$. I've proved that ...
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1answer
25 views

Given a set D = {a+b•| a,b ∈ $\mathbb{R}$} and a made-up binary operation, in a quadratic equation.

Given a set D = {a+b•| a,b ∈ $\mathbb{R}$} And a made-up binary operation on D is defined as follows: (a+b•)(c+d•)= ac+(ad+bc)• For example, (2+3•)(-3+5•)= (-6+1•) You are not allowed to combine ...
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1answer
63 views

Mechanics - Homework

A stone is thrown vertically down from a high building with an initial velocity of $4\;\mathrm ms^{-1}$. Calculate the time required for the stone to travel $30\;\mathrm m$. So far I have tried using ...
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6answers
241 views

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root $a,b$ are real numbers. Roots may or may not be real. In this question, the aim is to find values of and b ...
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3answers
80 views

Finding value of equation without solving for a quadratic equation

How do I go about solving this problem: If $α$ and $β$ are the roots of $x^2+2x-3=0$, without solving the equation, find the values of $α^6 +β^6$. In my thoughts: I commenced by expanding $(α ...
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1answer
32 views

Finding number of solutions.

How many solutions does this equation have $$2 \cos^2\left(\frac12 x \right) \sin^2 x = x^2+x-2$$ where $0 \lt x \le \displaystyle\frac \pi9?$ I observed that $2 \cos^2\left(\frac12x\right)$ can be ...
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1answer
51 views

Quadratic Equations - Mixed Roots

This is probably a silly question but why is it that when a quadratic equation has a single root it must be a repeated root. Why can't the second root be an imaginary root?
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3answers
319 views

A new way of solving cubics?

I found this (from http://www.quora.com/Mathematics/What-are-some-interesting-lesser-known-uses-of-the-quadratic-formula): So my question is: Can this be generalized to solve any depressed cubic ...
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1answer
57 views

if f(x) is the polynomial (coeff of leadin term is unity) in 'x' of least degree such that f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1, then f(0)=?

If $f(x)$ is the polynomial (coefficient of leading term is unity) in 'x' of least degree such that $f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1$ Then $f(0)= ?$
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2answers
121 views

How to prove that if $-1<x<0$ then $x^2 + x < 0$?

I am trying to prove an equivalence. I have already proved that: $$x^2 + x < 0 \implies -1 < x < 0 $$ using a sub-proof by cases, in which I used the fact that when $xy < 0$, $x$ and ...
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2answers
474 views

How do I determine the maximum value for a quadratic equation on an interval?

I need to determine the maximum value for y = ax^2 + bx + c, where I know the coefficients and the upper and lower x values. Say the input values are: a = 5 b = ...
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1answer
26 views

absolute value in a quadratic

If $a<-2$ is a real number, then the equation: $x^2+a|x|+1=0$ has how many real roots? After finding the roots in terms of $a$, how do I proceed?
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1answer
113 views

Finding a+b+c+d where a,b and c,d are the roots of two different quadratic equations

If $a, b$ are the roots of the equation $x^2-10cx-11d=0$ and $c,d$ are the roots of the equation $x^2-10ax-11b=0$ (where $a\ne b\ne c\ne d\ne 0$), then find the value of $a+b+c+d$. I have the ...
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1answer
36 views

Quadratic Equations and graphs [closed]

A bridge forms a parabolic arch. The span of the arch is 80 meters and its centre is 15 meters above either end. Write a quadratic equation that models the arch.
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1answer
65 views

Modular quadratic equation (solve for 3-digit natural numbers)

$n^2 + 6n - 88$ is divisible by 97. Solve for all n if n is a 3-digit natural number. Here's my progress so far $$n^2 + 6n - 88 \equiv 0\pmod {97}$$ $$n^2 + 6n - 88 + 97 \equiv 0\pmod {97}$$ $$n^2 + ...
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0answers
207 views

Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
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1answer
56 views

Algebra formulas: answer is positive, but in calculator it's negative.

$$-X^2 + 11X - 30 = 0 $$ $$\frac{-11 + \sqrt{11^2 -4 * 1*30}}{2*1} => \frac{-11 + \sqrt{1}}{2} => -5$$ Why do I get minus? In the book, it shows 5, not -5?
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2answers
37 views

How to factorize this quadratic?

How do i factorize this equation: $a(b-c)x^2 + b(c-a)x + c(b-a) = 0$ I tried the quadratic formula, but the discriminant is not factorising into a perfect square. Please help!
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2answers
84 views

Pythagoras numbers and fermats last theorem

I am reading "What Is Mathematics? An Elementary Approach to Ideas and Methods" And I am stuck here, I don't get it. I have posted a screen shot underlining what my doubt is.. I dont get it when the ...
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3answers
110 views

If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist?

If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist? One Two Three Infinitely many Also, what are these pairs?
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2answers
94 views

Proving the second root of a quadratic equation

If $\alpha$ is a root of the equation $4x^2+2x-1=0$, then prove that $4\alpha^3-3\alpha$ is the other root. How do I proceed? The sum of the roots, the product of the roots lead me nowhere. Should I ...
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1answer
441 views

Symmetric System of Equations

I'm new on studying Systems of equations. I just want to know the number of real solutions of this system of equations: \begin{align*} x^2-y^2=z\\ y^2-z^2=x\\ z^2-x^2=y \end{align*} I also want to ...
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4answers
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Roots of $x^2+3x+2=0$ are infinite !!! [closed]

I have quadratic equation here: $x^2+3x+2=0$ so $(x+2)(x+1)=0$ and I can do $(x+2)=0/(x+1)$ and that solution of the equation is $x+2=0$ so $x=-2$ but my teacher said that it is wrong why? ...
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3answers
42 views

Find the integer n

Let $a$ and $b$ be two integers such that $10a+b=5$ and $p(x)=x^2+ax+b$. Find the integer $n$ such that $p(10)p(11)=p(n).$ Please tell how to proceed.
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1answer
58 views

Proving the minimum value of (x+a)(x+b)/(x+c)

Show that the minimum value of $\frac {(x+a)(x+b)}{(x+c)}$, where a$\gt$c, b$\gt$c, is $(\sqrt{a-c}+\sqrt{b-c})^{2}$ for real values of x$\gt-c$. I did $$\frac {(x+a)(x+b)}{(x+c)}=y$$ and then took ...
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2answers
70 views

Proper method for solving quadratic equations with exponents

$(\sqrt {x^2-5x+6}+\sqrt{x^2-5x+4})^{x/2}$ + $(\sqrt {x^2-5x+6}-\sqrt{x^2-5x+4})^{x/2}$ = $2^{(x+4)/4}$ I have found out, by trial and error method, that $x=0$ and $x=4$ satisfy this equation. But is ...
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3answers
107 views

Prove that for real numbers $x$, if $x^2 - 5x + 4 \ge 0$, then either $x \le 1$ or $x \ge 4$.

Its another homework question that I'm having trouble understanding. The full question is write a detailed structured proof that uses a proof by cases to prove that for real numbers $x$, if $x^2 - 5x ...
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2answers
62 views

Quadratic inequalities

This is what I tried. I tried finding limits of y and then equating them with the given limits, but I could not simplify it further. The given options for this question are: a+b=23 a^2+b^2=277 ...
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3answers
90 views

Quadratic equation problems

There is a circle with a radius of $25$ ft and origin at $(0, 0)$ and a line segment from (0, -31) to (-37, 8). Find the intersections of the line and circle. I am asking for somebody to analyze ...
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1answer
343 views

Probabilistic Robotics Exercise

I am reading Probabilistic Robotics and I don't know how to solve the exercise problem number 4 at the end of the second chapter. There are no solutions to this text. The exercise states: In ...
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2answers
102 views

Why/when did these extraneous solutions appear while solving a quadratic equation?

I am trying to solve the quadratic equation $x^2 + x + 1 = 0$. $x^2 = -1 - x $ $\iff x = -\frac{1}{x} - 1$, assuming $x\neq 0$. Substituting that into the original equation gives $x^2 + (-\frac{1}{x} ...
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1answer
24 views

Find a if the equation has a solution

If this equation has a solution, then 'a' is equal to None of these How should I proceed in this problem?
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2answers
38 views

Least Value of the Quadratic Expression

What is the least value of this expression? Please show me a way to determine it.
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2answers
462 views

Factoring Quadratics: Asterisk Method

I'm teaching my students about factoring quadratics. We've done GCF, difference of two squares, squared binomials, and grouping. One of my colleagues then found this asterisk method on line. It's ...
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2answers
225 views

Solving system of multivariable 2nd-degree polynomials

How would you go about solving a problem such as: \begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix} where $(x,y)\in\mathbb{C}^{2}$. More generally, how would ...
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4answers
616 views

Real world examples of quadratic and/or finding roots of a quadratic?

Anyone ever come across a good situation where a) a situation is modeled by a quadratic equation $y=ax^2+bx+c$ and/or b) you've even needed to find where $y=0$ (roots, $x$-intercept, etc)
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1answer
538 views

by completing the square find in terms of k the roots of the equation $x^2 + 2kx-7=0$

By completing the square find in terms of $k$ the roots of the equation $$x^2 + 2kx-7=0$$ prove for all real values of $k$, the roots are real
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4answers
292 views

Derivation of the quadratic equation

So everyone knows that when $ax^2+bx+c=0$,$$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ But why does it equal this? I learned this in maths not 2 weeks ago and it makes no sense to me
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1answer
144 views

Why does completing the square give you the minimum point?

Say we have an equation:$y=$ ${x^2} + 2x + 1$ Completing the square we get: $\eqalign{ & y={x^2} + 2x + 1 \cr & = {(x + 1)^2} - {(1)^2} + 1 \cr & = {(x + 1)^2} \cr} $ The ...
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1answer
41 views

Algebra steps breakdown

I have been reviewing this question. : How can I find the points at which two circles intersect? Although during the answering steps I got a little stuck: From this: 17y2−62y+49=0 To this: ...
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1answer
331 views

Finding integral roots of $x^2 + px + q = 0$ if $p+q=198$.

Given the relation that $p+q=198$, the question is to find all the integral roots of the equation: $$ x^2+px + q = 0 $$ How to proceed? I know we'll have to use Vieta's formulas, but I don't know ...
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1answer
60 views

$\frac{\text{Quadratic}}{\text{Quadratic}}$ methods

So, lets take up a question. Show that $f(x)=\frac{x^2+34x-71}{x^2+2x-7}$ can never lie between 5 and 9. MY ATTEMPT: I assumed the function to be equal to k , then cross multiplied and got a single ...
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1answer
33 views

One way to describe the pattern of covariation for a linear function is:

One way to describe the pattern of covariation for a linear function is: As input value increases by 1, the output value changes by a constant (fixed) amount k where k is some real number. Explain why ...
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2answers
1k views

Finding vertex and focus of parabola given an equation

I am defeated to complete square on the following parabolic equation. Please help. Find the vertex and focal width for the parabola: $$ x^2+6x+8y+1=0 $$ I am hoping to get an equation in this form ...
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1answer
129 views

Find conditions on a,b,c so that p(x) and q(x) have exactly 2 roots in common. Also solve the equation p(x)=0

Let $p(x) = x^4 +ax^3 +bx^2+cx +1$ and $q(x) = x^4 +cx^3 +bx^2+ax+1$ with a,b,c real numbers.Find conditions on a,b,c so that p(x) and q(x) have exactly 2 roots in common. Also solve the equation ...
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1answer
52 views

Solving Quadratics

A rectangular lawn measures 30 m by 40 m. Jason is cutting the lawn from the outside perimeter in toward the centre by cutting strips along the entire perimeter first, then continuing as he cuts ...
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2answers
69 views

Solving $y^2(x^2+1) +x^2(y^2+16) =448$

$y^2(x^2+1) +x^2(y^2+16) =448$ The task is to find all solutions in integers $(x,y)$. This is the fourth question of rmo 1st stage.The solution here is not complete. I have tried to solve unable to. ...
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1answer
70 views

Solving a quadratic made from the sum of monomial denominators. [closed]

Solve the following equation. Separate your answers with commas. Repeated roots should only be entered once. $$\frac{1}{x-5} + \frac{1}{x-6} = \frac{11}{30}$$ Any ideas on how to start out?
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1answer
96 views

sub rectangle region from combined area

I have a rectangle divided like so ...
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2answers
48 views

Confusion with qudratic equations

According to my book In the given equation $$x^2+x+1=0\tag{1}$$ If $a$ is a root of eqn $(1)$ then $a$ satisfies the following equation $$a^2+a+1=0\tag{2}$$ $$\implies ...