# 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 ...
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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 ...
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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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### Let $f(x)=x^2+12x+30$. Solve $f(f(f(f(f(x)))))=0$

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 ...
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### 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 ...
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 $$z_0 = ... 3answers 3k views ### How to solve equations to the fourth power? Is it possible to manually retrieve the value of y from the following equation$$\color{blue}{153y^2-y^4=1296}$$WolframAlpha has four solutions for y: -12, -3, 3, 12. How has it solved? What ... 1answer 354 views ### Nested solutions of a quadratic equation. A quadratic equation of the form x^2+bx+c=0 can be solved with the classical formula that gives all solutions. Here I want discuss some other methods to find one solution. The best known is by ... 8answers 426 views ### What is the connection between the discriminant of a quadratic and the distance formula? The x-coordinate of the center of a parabola ax^2 + bx + c is$$-\frac{b}{2a}$$If we look at the quadratic formula$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$we can see that it specifies two ... 10answers 4k views ### Taking Calculus in a few days and I still don't know how to factorize quadratics Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ... 2answers 263 views ### Find all pair of cubic equations Find all pair of cubic equations x^3+ax^2+bx+c=0 and x^3+bx^2+ax+c=0, where a,b are positive integers and c not equal to 0 is an integer, such that both the equations have three integer ... 1answer 281 views ### Convergence of the quadratic map \left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2? Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for ... 1answer 85 views ### Aside from the obvious stuff, do the partial functions that solve the quadratic equation have any interesting properties? Let us define partial functions$$f_+,f_- : \mathbb{R} \leftarrow \mathbb{R} \times \mathbb{R} \times \mathbb{R}$$so as to return the zeros of the quadratic ax^2+bx+c whenever they exist, such ... 1answer 203 views ### On the prime-generating polynomial m^2+m+234505015943235329417 In 2009, J. Waldvogel and Peter Leikauf found the remarkable Euler-like polynomial,$$F(m)=m^2+m+234505015943235329417$$which is prime for m=0\to20, but composite for m=21. Define, ... 8answers 644 views ### How to factor quadratic ax^2+bx+c? How do I shorten this? How do I have to think?$$ x^2 + x - 2$$The answer is$$(x+2)(x-1)$$I don't know how to get to the answer systematically. Could someone explain? Does anyone have a link to ... 6answers 1k views ### If 3x^2 -2x+7=0 then (x-\frac{1}{3})^2 =? If 3x^2-2x+7=0 then$$\left(x-\frac{1}{3}\right)^2 =\text{?}$$I am so confused. It is a self taught algebra book. The answer is \large -\frac{20}{9} but I don't know how it was derived. ... 3answers 3k views ### How many fingers do martians have? Text of problem: It is supposed that we use base 10 as our number system because we have ten fingers. A martian, after seeing the equation x^2-16x+41=0 writes the difference of the roots as 10. ... 6answers 698 views ### Is x^{\frac{1}{2}}+ 2x+3=0 a quadratic equation Is$$x^{\frac{1}{2}}+ 2x+3=0$$considered a quadratic equation? Should the equation be in the form$$ax^2+bx+c=0$$to be considered quadratic? 3answers 369 views ### A new way of solving cubics? I found this (from http://www.quora.com/Mathematics/What-are-some-interesting-lesser-known-uses-of-the-quadratic-formula): So my question is: Can this be generalized to solve any depressed cubic ... 6answers 1k views ### what would be the way to solve a system of equations like this one? Solve: xy=-30 x+y=13 {15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ? 3answers 524 views ### Quadratic Formula, nature of roots with Trigonometric Functions The original problem: If 0\le a,b\le 3 and the equation$$x^2+4+3\cos(ax+b)=2x$$has at least one real solution, then find the value of a+b$$$$At first, on rearranging, I got the ... 2answers 123 views ### f(g(h(x)))=0 has 8 real roots Find all quadratic polynomials f(x),g(x) and h(x) such that the polynomial f(g(h(x)))=0 has roots 1,2,3,4,5,6,7 and 8. I don't know what to do. Making a 8 degree equation is quite ... 1answer 113 views ### Replacing numbers by roots of quadratic We have 10 numbers in the interval (0,1), not necessarily distinct. At any moment, we can choose two of them, a and b. If the quadratic x^2-ax+b has two (possibly identical) real roots, we ... 5answers 261 views ### Solving 2^{2x+1} - 2^{x+4} = 2^3 - 2^x$$2^{2x+1} - 2^{x+4} = 2^3 - 2^x$$How can I solve an exponential equation that has many terms as the one above. Include more than one method if available. 4answers 804 views ### Simple Trig Equations - Why is it Wrong to Cancel Trig Terms? In the following problem, I first did it using a cancellation of sin^2\theta, working shown below, which gave the wrong answer. Having looked at the question again, I saw it could be solved by ... 4answers 180 views ### Find the solution of \lfloor{x^2}\rfloor−\lfloor{3x}\rfloor+2=0 Is anyone able to help me with the following equation concerned the floor function \lfloor{x^2}\rfloor−\lfloor{3x}\rfloor+2=0 I don't know how to deal with the floor terms properly. 2answers 101 views ### p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}; \sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}} = ?  Let p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}} The sum$$\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}}$$Can be expressed as \frac{a}{b}. Where \lfloor \cdot \rceil denotes the ... 3answers 307 views ### Quadratic Equation Recurrence? So I was playing around with the following: Let f_0(x) = x^2 - bx + c. If f_n(x) has roots p and q with p > q, then let f_{n+1}(x) = x^2 - px + q. The recurrence relation is rather ... 1answer 140 views ### Coincidence? : d(ax^2+bx+c)/dx=\pm \sqrt{\Delta} As the title says, is it just a coincidence that d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}? (where \Delta=b^2-4ac, i.e. discriminant of the quadratic). We can get this easily from rearranging the ... 4answers 382 views ### Why are there four solutions to x^2-2x-8=0 in \mathbb{R}? Or am I wrong? It might be a very trivial question to ask but why do we get four different solutions for a quadratic equation using these two methods? x^2-2x-8=0 We see that factors are (x-4) and (x+2) so ... 3answers 265 views ### Compute an expression without calculating the roots Let a and b be the roots of this equation: x^2 - x - 5 = 0 Find the value of (a^2 + 4b - 1)(b^2 + 4a - 1) Without ... 5answers 238 views ### How to solve \ x^2-19\lfloor x\rfloor+88=0  I have no clue on how to solve this. If you guys have, please show me your solution as well.$$\ x^2-19\lfloor x\rfloor+88=0 $$4answers 215 views ### Both solutions to a quadratic make sense — looking for applications I'm looking for reasonably real, non-abstract applications modeled by quadratic equations where both solutions make sense. I'd like them to be accessible to high school algebra students. One I come ... 4answers 729 views ### Exponential Simultaneous Equations Solve the following simultaneous equations:$$2^x + 2^y = 10x + 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 ... 3answers 761 views ### Universal quadratic formula? 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 = ... 6answers 414 views ### 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 ... 1answer 176 views ### Geometric derivation of the quadratic equation The quadratic equation can be thought of as specifying distances in the Euclidean plane. It tells us that the x-intercepts of a function occur at a distance of \frac{\sqrt{b^2-4ac}}{2a} from the ... 2answers 197 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 ... 3answers 118 views ### how to calculate roots of given equation below? Without solving equation 2x^2 + 9x + 9 = 0, show that one of the root of the equation is twice the other. 1answer 89 views ### Prove that n is divisible by 6 Problem: Let x^2+mx+n and x^2+mx-n give integer roots where (m,n) are integers. Show that n is divisible by 6 My attempt: Since the roots are integers then the discriminants of both the ... 2answers 64 views ### Find the values of \cos(\alpha+\beta)  if the roots of an equation are given in terms of tan It is given that  \tan\frac{\alpha}{2}  and  \tan\frac{\beta}{2}  are the zeroes of the equation  8x^2-26x+15=0 then find the value of \cos(\alpha+\beta). I attempted to solve this but I ... 8answers 941 views ### Solving the following trigonometric equation: \sin x + \cos x = \frac{1}{3}  I have to solve the following equation:$$\sin x + \cos x = \dfrac{1}{3} $$I use the following substitution:$$\sin^2 x + \cos^2 x = 1 \longrightarrow \sin x = \sqrt{1-\cos^2 x} And by ...
A quadratic polynomial $p(x)$ is such that $p(x)$ never takes any negative values. Also, $p(0)=8$ and $p(8)=0$. What would $p(-4)$ be? I tried doing it by taking the minimum value as zero that is ...