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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0answers
74 views

Quadratic equations with prime coefficients

I recently decided to go through old high school notebooks and I found something marginally interesting. I used to note down all kinds of things I came across, and I thought this might be useful for ...
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1answer
94 views

To find ? in equation $a^2+?^2=c$

How can we solve for $?$ in the below given Equation: $$a^2+?^2=c$$ I don§t want to use Square or Square root as the the number can be in decimals.
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2answers
72 views

Complex numbers - Quadratic formula?

Let a and b be real numbers. The complex number 4 - 5i is a root of the quadratic $z^2 + (a + 8i) z + (-39 + bi) = 0$. What is the other root? I did a lot of work on hand and plugging this into the ...
0
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2answers
81 views

Query about a statement on the consequence of two quadratic equations having a common root

I have read an answer (in this site, but I lost the question number) saying something like the following:- If the quadratic equations F(x) = 0 and f(x) = 0 have a common root, then the quadratics are ...
3
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5answers
1k views

Condition for a common root in two given quadratic equations

If $a,\;b,\;c$ are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root if $\;\displaystyle\frac da,\;\frac eb,\;\frac fc$ are in: Arithmetic Progression ...
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2answers
31 views

Find quadratic equation based on 2 tangents

I would like to know a way to find an quadratic equation that had 2 given tangents: For example here is 2 tangents equations: y = 1/2 x y = 2 x + 2 and 2 abscisses x = 0 x = 3 Is there a ...
0
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3answers
52 views

Solving an equation involving $x^2$

I have come to a question with the equation: $$6 = x^2 -7x + 6.$$ The answer is $7$. How do I do I find the solution to a problem involving $x^2$?
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3answers
361 views

Solving a Quadratic Equation “Using a Table and a Graph”

I need to find $a\in \Bbb Z, 0\le a\lt10 : f(1 + \frac{a}{10}) = 0$ for a number of different quadratic functions, for example $f(x) = -x^2 + 4x - 3$, by "using a table and a graph". Can someone ...
0
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1answer
36 views

information content of a quadratic surd

how much information is required to construct the equation: $$ X^2 - 2=0 \; ? $$ suppose, in a spirit of seasonal festivity, we squander a few further bits, and pamper ourselves with the additional ...
0
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0answers
60 views

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then $A.$ a,b,c,d are in G.P. $B.$ a,b,c,d are in H.P. $C.$ a,b,c,d are in A.P. $D.$ None of the above Tried writing the expression as a ...
2
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1answer
55 views

how to prove roots quadratics

the quadratic equation $3(k+2)x^2+(k+5)x+k=0$ has real roots show $(k-1)(11k+25) \geq 0 $ If $\Delta$ greater than $0$ it has real roots so, $$\Delta = (k+5)^2 - 4 \cdot (3(k+2))\cdot k$$ ...
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5answers
69 views

$x^2+y^2=1, 5x+12y+13=0$ Simultaneous Equations

Can someone solve this for me and show working out? I just can't do it and I don't know why I am getting x and y wrong. It will be very much appreciated. As basic as possible as well please.
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1answer
112 views

Finding two unknowns in two quadratic polynomials with only knowing the divisors

There are two quadratic polynomials (dividends). These two polynomials are divided by two different linear polynomials like $x+1$ (divisors). The remainders are known, but the quotients are unknown. ...
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1answer
51 views

How can I solve $\frac{2x}{\sqrt{1-x^2}}=0$

This is what I could come up with: $\dfrac{2x}{\sqrt{1-x^2}}=0$ $\left(\dfrac{2x}{\sqrt{1-x^2}}\right)^2=0^2$ $\dfrac{4x^2}{1-x^2}=0$ I can't go forward from this point because of that stupid ...
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0answers
15 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
1
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1answer
44 views

“Alternative factorising method” for quadratics not working

In class my teacher showed us an alternative method for factorising quadratics which are more awkward (i.e. the $a$ in $ax^2+bx+c$ is greater than 1). The method is: 1. Take your quadratic (e.g. ...
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0answers
58 views

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 ...
0
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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 ...
2
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1answer
57 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
231 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
75 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 ...
3
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1answer
29 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
36 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
295 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 ...
0
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1answer
52 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 $f(0)= ?$
2
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2answers
98 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
271 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
21 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
91 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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0answers
39 views

Complete the square

How would I complete the square for $y$ after completing the square for x below. Note that $y,x$ are vectors and not scalars. $$ (y-A^Tx)^TC(y-A^Tx)+(x-\mu)^TD(x-\mu)\\ ...
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1answer
35 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
48 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 + ...
3
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0answers
162 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
55 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
31 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!
2
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2answers
76 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 ...
2
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3answers
102 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
86 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
358 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
164 views

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.
1
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1answer
49 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 ...
3
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2answers
68 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 ...
2
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3answers
99 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 ...
2
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2answers
56 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 ...
2
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3answers
88 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 ...
1
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1answer
247 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 ...
6
votes
2answers
88 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} ...
0
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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?
1
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2answers
36 views

Least Value of the Quadratic Expression

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