# Tagged Questions

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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### Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
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### Why do I get one extra wrong solution?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by (-1): $$\sqrt{x}=x-2$$ power of 2: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $1$ is not a ...
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### Why can we prove mathematically that a formula to solve an (n+5) order polynomial does not exist?

I understand that the quadratic equation can solve any second order polynomial. Furthermore, equations exist for polynomials up to fourth order. However, without a graduate level degree and a deep ...
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### Is it possible for a quadratic equation to have one rational root and one irrational root?

Is it possible for a quadratic equation to have one rational root and one irrational root? Yes, a pretty straightforward question. Is it possible?
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### I think I can complete the square of any quadratic, is it true? (Any reason to ever use Quad. Formula?)

I was taught that you could only complete the square of a quadratic if the coefficient on the $x^2$ term is 1. However, playing a little bit with other quadratics, I've found that it's just not true....
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### 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 ...
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### Solve $x^4+3x^3+6x+4=0$… easier way?

So I was playing around with solving polynomials last night and realized that I had no idea how to solve a polynomial with no rational roots, such as $$x^4+3x^3+6x+4=0$$ Using the rational roots test, ...
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### Probability of $ax^2 + bx + c = 0$ having real solutions

$a$, $b$, $c$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $ax^2 + bx + c = 0$ has real solutions? ...
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### An interesting table of Prime Generating polynomials similar to $n^2+n+41$?

Here is some data on quadratic prime-generating polynomials of a particular form. Kindly look at the questions given below it. Note: The discriminant $d$ is square-free and its class number $h(d)$ is ...
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### Why are equations written by equating something to zero?

A linear equation is $$ax + b = 0 ; \,\, \,\, a\neq 0$$ A quadratic equation is $$ax^2 + bx + c = 0 ; \,\, a\neq 0$$ And so on... Why are all these equations written as $\dots = 0$? Why do ...
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Here is my solve, is it correct? I figured out that we can restate $f(f(x))$ as $((x+r)(x+s)+r)((x+r)(x+s)+s)$ thus $f(f(f(f(f(x)))))=0$ is $(x+r)^2(x+s)^2(4s+3r)(4r+3s)$ from vieta's $0=(x+r)(... 4answers 1k views ### Find$C$such that$x^2 - 47x - C = 0$has integer roots, and further conditions Have been working on this for years. Need a system which proves that there exists a number$C$which has certain properties. I will give a specific example, but am looking for a system which could ... 3answers 439 views ### What would be the value of$\sum\limits_{n=0}^\infty \frac{1}{an^2+bn+c}$I would like to evaluate the sum $$\sum_{n=0}^\infty \frac{1}{an^2+bn+c}$$ Here is my attempt: Letting $$f(z)=\frac{1}{az^2+bz+c}$$ The poles of$f(z)$are located at $$z_0 = \frac{-b+\sqrt{... 3answers 3k views ### How to solve equations to the fourth power? Is it possible to manually retrieve the value of y from the following equation$$\color{blue}{153y^2-y^4=1296}$$WolframAlpha has four solutions for y: -12, -3, 3, 12. How has it solved? What ... 1answer 372 views ### Nested solutions of a quadratic equation. A quadratic equation of the form x^2+bx+c=0 can be solved with the classical formula that gives all solutions. Here I want discuss some other methods to find one solution. The best known is by ... 8answers 442 views ### What is the connection between the discriminant of a quadratic and the distance formula? The x-coordinate of the center of a parabola ax^2 + bx + c is$$-\frac{b}{2a}$$If we look at the quadratic formula$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$we can see that it specifies two ... 10answers 4k views ### Taking Calculus in a few days and I still don't know how to factorize quadratics Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ... 2answers 269 views ### Find all pair of cubic equations Find all pair of cubic equations x^3+ax^2+bx+c=0 and x^3+bx^2+ax+c=0, where a,b are positive integers and c not equal to 0 is an integer, such that both the equations have three integer ... 1answer 313 views ### Convergence of the quadratic map \left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2? Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the ... 1answer 87 views ### Aside from the obvious stuff, do the partial functions that solve the quadratic equation have any interesting properties? Let us define partial functions$$f_+,f_- : \mathbb{R} \leftarrow \mathbb{R} \times \mathbb{R} \times \mathbb{R}$$so as to return the zeros of the quadratic ax^2+bx+c whenever they exist, such ... 1answer 205 views ### On the prime-generating polynomial m^2+m+234505015943235329417 In 2009, J. Waldvogel and Peter Leikauf found the remarkable Euler-like polynomial,$$F(m)=m^2+m+234505015943235329417$$which is prime for m=0\to20, but composite for m=21. Define,$$F(m)=m^2+... 8answers 646 views ### How to factor quadratic$ax^2+bx+c$? How do I shorten this? How do I have to think? $$x^2 + x - 2$$ The answer is $$(x+2)(x-1)$$ I don't know how to get to the answer systematically. Could someone explain? Does anyone have a link to ... 6answers 2k views ### If$3x^2 -2x+7=0$then$(x-\frac{1}{3})^2 =$? If$3x^2-2x+7=0$then $$\left(x-\frac{1}{3}\right)^2 =\text{?}$$ I am so confused. It is a self taught algebra book. The answer is$\large -\frac{20}{9}$but I don't know how it was derived. ... 3answers 3k views ### How many fingers do martians have? Text of problem: It is supposed that we use base 10 as our number system because we have ten fingers. A martian, after seeing the equation$x^2-16x+41=0$writes the difference of the roots as$10$. ... 6answers 723 views ### Is$x^{\frac{1}{2}}+ 2x+3=0$a quadratic equation Is $$x^{\frac{1}{2}}+ 2x+3=0$$ considered a quadratic equation? Should the equation be in the form $$ax^2+bx+c=0$$ to be considered quadratic? 3answers 370 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 [... 6answers 1k views ### what would be the way to solve a system of equations like this one? Solve:$xy=-30x+y=13${15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ? 3answers 531 views ### Quadratic Formula, nature of roots with Trigonometric Functions The original problem: If$0\le a,b\le 3$and the equation $$x^2+4+3\cos(ax+b)=2x$$ has at least one real solution, then find the value of$a+b$ At first, on rearranging, I got the ... 2answers 133 views ###$f(g(h(x)))=0$has$8$real roots Find all quadratic polynomials$f(x),g(x)$and$h(x)$such that the polynomial$f(g(h(x)))=0$has roots$1,2,3,4,5,6,7$and$8$. I don't know what to do. Making a$8$degree equation is quite tedious.... 1answer 55 views ### If$a$and$b$be the roots of the quadratic equation$x^2-6x+4=0$then find the value of given expression. Let$a$and$b$be the roots of the quadratic equation$x^2-6x+4=0$and$P_n = a^n + b^n$then the value of $$\frac{P_{50}(P_{48}+P_{49})-6P_{49}^2+4P_{48}^2}{P_{48}.P_{49}}$$ Options are$(A)2$... 1answer 115 views ### Replacing numbers by roots of quadratic We have$10$numbers in the interval$(0,1)$, not necessarily distinct. At any moment, we can choose two of them,$a$and$b$. If the quadratic$x^2-ax+b$has two (possibly identical) real roots, we ... 5answers 263 views ### Solving$2^{2x+1} - 2^{x+4} = 2^3 - 2^x$$$2^{2x+1} - 2^{x+4} = 2^3 - 2^x$$ How can I solve an exponential equation that has many terms as the one above. Include more than one method if available. 4answers 831 views ### Simple Trig Equations - Why is it Wrong to Cancel Trig Terms? In the following problem, I first did it using a cancellation of$sin^2\theta$, working shown below, which gave the wrong answer. Having looked at the question again, I saw it could be solved by ... 4answers 188 views ### Find the solution of$\lfloor{x^2}\rfloor−\lfloor{3x}\rfloor+2=0$Is anyone able to help me with the following equation concerned the floor function$\lfloor{x^2}\rfloor−\lfloor{3x}\rfloor+2=0$I don't know how to deal with the floor terms properly. 2answers 103 views ###$p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$;$\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}} = ? $Let$p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$The sum $$\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}}$$ Can be expressed as$\frac{a}{b}$. Where$\lfloor \cdot \rceil$denotes the nearest ... 2answers 199 views ### Find the number of sets of$(a,b,c)$for$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{29}{72}$If$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{29}{72},\ \ c<b<a<60,\ \ \{a,b,c\}\in\mathbb{N} $. How many sets of$(a,b,c)$exists ? Options$a.)\ 3 \quad \quad \quad \quad \quad ...
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So I was playing around with the following: Let $f_0(x) = x^2 - bx + c$. If $f_n(x)$ has roots $p$ and $q$ with $p > q$, then let $f_{n+1}(x) = x^2 - px + q$. The recurrence relation is rather ...
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### Coincidence? : $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$

As the title says, is it just a coincidence that $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$? (where $\Delta=b^2-4ac$, i.e. discriminant of the quadratic). We can get this easily from rearranging the ...
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### Why are there four solutions to $x^2-2x-8=0$ in $\mathbb{R}$? Or am I wrong?

It might be a very trivial question to ask but why do we get four different solutions for a quadratic equation using these two methods? $x^2-2x-8=0$ We see that factors are $(x-4)$ and $(x+2)$ so ...
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### Compute an expression without calculating the roots

Let a and b be the roots of this equation: x^2 - x - 5 = 0 Find the value of (a^2 + 4b - 1)(b^2 + 4a - 1) Without ...
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### How to solve $\ x^2-19\lfloor x\rfloor+88=0$

I have no clue on how to solve this. If you guys have, please show me your solution as well. $$\ x^2-19\lfloor x\rfloor+88=0$$
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### Both solutions to a quadratic make sense — looking for applications

I'm looking for reasonably real, non-abstract applications modeled by quadratic equations where both solutions make sense. I'd like them to be accessible to high school algebra students. One I come ...
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### Exponential Simultaneous Equations

Solve the following simultaneous equations: $$2^x + 2^y = 10$$ $$x + y = 4$$ Looking at it, it is obvious that the answers are $(3,1)$ and $(1,3)$, however, I was wondering if they could be solved ...
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### How to solve the following? $x^3+1=2{(2x-1)}^{1/3}$.

Find all the real solutions of $$x^3+1=2{(2x-1)}^{1/3}$$ I tried to cube both sides but got messed up with a nine degree equation! Please help. Thanks in advance!
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Is there any way to write the quadratic formula such that it works for $ac= 0$ without having to make it piecewise? The traditional solution of $x = (-b \pm \sqrt{b^2 - 4ac}) / 2a$ breaks when $a = 0$...
### Solve $5a^2 - 4ab - b^2 + 9 = 0$, $- 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0$
Solve $\left\{\begin{matrix} 5a^2 - 4ab - b^2 + 9 = 0\\ - 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0. \end{matrix}\right.$ I know that we can use quadratic equation twice, but then we'll get some very ...