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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138
votes
18answers
14k views

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 ...
35
votes
4answers
2k views

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 ...
25
votes
4answers
6k views

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?
22
votes
5answers
3k views

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 ...
21
votes
1answer
210 views

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, ...
20
votes
10answers
3k views

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 ...
19
votes
5answers
330 views

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 ...
17
votes
3answers
418 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 = ...
17
votes
2answers
330 views

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. $$\begin{array}{cccc} \text{#} & P(n)=an^2+bn+c\,; & d = ...
16
votes
2answers
307 views

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 ...
15
votes
8answers
293 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 ...
13
votes
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 ...
11
votes
8answers
623 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 ...
10
votes
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" ?
10
votes
3answers
337 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 ...
9
votes
5answers
541 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?
9
votes
3answers
233 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 ...
9
votes
1answer
101 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 ...
8
votes
5answers
238 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.
7
votes
4answers
306 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 ...
7
votes
3answers
262 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 ...
7
votes
4answers
213 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 ...
7
votes
3answers
618 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 = ...
7
votes
6answers
396 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 ...
7
votes
3answers
97 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.
7
votes
1answer
119 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 ...
7
votes
1answer
81 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 ...
6
votes
2answers
554 views

If p(x) is a non-negative quadratic polynomial, p(0)=8 and p(8)=0, what is p(-4)?

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 ...
6
votes
9answers
3k views

Prove $ax^2+bx+c=0$ has no rational roots if $a,b,c$ are odd

If $a,b,c$ are odd, how can we prove that $ax^2+bx+c=0$ has no rational roots? I was unable to proceed beyond this: Roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and rational numbers are of the form ...
6
votes
4answers
199 views

Solving an exponential equation involving e: $e^x-e^{-x}=\frac{3}{2}$

In a previous exam, my professor had the question \begin{equation*} e^x-e^{-x}=\frac{3}{2}. \end{equation*} I attempted to take the natural log of both side to solve it, but evidently that was ...
6
votes
2answers
351 views

quadratic equation what am I doing wrong?

solve $$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} $$ $$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} \Rightarrow $$ $$ 5x+19 = (x+7) + 4\sqrt{x-5}\sqrt{x+7} + (x+5) \Rightarrow $$ $$ 3x + 17 = ...
6
votes
3answers
131 views

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!
6
votes
1answer
317 views

Why is the '+' solution to the quadratic formula always the one that satisfies my constraint

Let $A,B \in (0,1)$ be known constants, and $C \in (-\infty, \infty)$ be a known constant. Define \begin{equation} \xi(x) = \log \big( x \big) + \log \big( 1+x-A-B \big) - \log \big( A - x \big) ...
6
votes
2answers
95 views

Solutions to $(4x^2+\frac{16}3x)^{\sqrt {3-x}}=(4x^2+\frac{16}3x)^{\sqrt {2x+11}-\sqrt{x+2}}$

$$(4x^2+\frac{16}3x)^{\sqrt {3-x}}=(4x^2+\frac{16}3x)^{\sqrt {2x+11}-\sqrt{x+2}}$$ I found the solutions to be $0, -\frac32, -1, -\frac43$ I can't figure out why any of those wouldn't work, but my ...
6
votes
4answers
770 views

For what values of m are the roots of $x^2 +2x+3 = m(2x+1)$ real and positive

I am only able to show that to be real, $m <-1$ or $m\geq2$ Don't know how to finish solution Answer is $2 \leq m < 3$ So far: After expanding and factorising, $x^2 + 2(1-m)x + (3-m) = 0 $ ...
6
votes
2answers
128 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} ...
5
votes
4answers
313 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
5
votes
5answers
7k views

Why a quadratic equations always equals zero?

On evaluating quadratic equations, It always equals zero: $$ax^2+bx+c=0$$ Why zero? Is it possible to use other number for another purpose?
5
votes
3answers
618 views

Solution to quadratic question of the form 0/0

What are the possible values of $x$ for the following equation: $$\frac{x - 1}{1 - x} = \frac1x$$ This equation is equivalent to $$x^2 - 1 = 0$$ which factors to $1, -1$. However, is $1$ the ...
5
votes
3answers
196 views

If $a+b=x$ and $ab=y$, what is the quickest way to solve for $a$ and $b$?

The mechanistic approach would be to simply substitute $b=y/a$ in the first equation to obtain a quadratic in $a$. But seeing the simplicity of the givens, I feel that there must be some better and ...
5
votes
2answers
472 views

How to solve this equation? Can I treat as a quadratic equation?

$$\ln(x+3)+\ln(x-4)=0$$ How to solve this equation? First removing the 'ln' from the equation and after making a quadratic equation and then solve the quadratic equation?
5
votes
4answers
147 views

How to solve an equation with $x^4$?

Today, I had this question on a Maths test about Algebra. This was the equation I had to solve: $$(1-x)(x-5)^3=x-1$$ I worked away the brackets and subtracted $x-1$ from both sides and was left with ...
5
votes
2answers
1k views

Solution af a system of 2 quadratic equations

I have a system of two quadratic equations with unknowns $x$ and $y$: $$a_{1 1} x y + a_{1 2} x^2 + a_{1 3} y^2 + a_{1 4} x + a_{1 5} y + a_{1 6} = 0,\\ a_{2 1} x y + a_{2 2} x^2 + a_{2 3} y^2 + a_{2 ...
5
votes
4answers
873 views

find the least a, for which two equations have a common root

Could you help me out please. I have two equations: $2x^2-3x+1=0 $ and $ 2x^2-(a+3)x+3a=0$ I need to find the least $a$ for which these two equations have a common root. At a first glance I thought ...
5
votes
4answers
176 views

Solve $x^{3}-3x=\sqrt{x+2}$

Solve for real $x$ $$x^{3}-3x=\sqrt{x+2}$$ By inspection, $x=2$ is a root of this equation. So, I squared both sides and divided the six degree polynomial obtained by $x-2$. Then I got a ...
5
votes
2answers
165 views

Find the value of $x_1^6 +x_2^6$ of this quadratic equation without solving it

I got this question for homework and I've never seen anything similar to it. Solve for $x_1^6+x_2^6$ for the following quadratic equation where $x_1$ and $x_2$ are the two real roots and $x_1 > ...
5
votes
2answers
769 views

Find the value of a + b + c + d

Let $a$ and $b$ be the roots of the equation: $x^2 - 10cx - 11d = 0$ where $c$ and $d$ be the roots of $x^2 - 10ax - 11b = 0$. Find the value of $a+b+c+d$, assuming that they all are distinct. I ...
5
votes
3answers
241 views

What are a , b and c?

$$y = ax^2 + bx + c$$ which is tangent at the origin with the line $y=x$, It is also tangential with the line $y=2x + 3$. Determine the function! Draw a figure! My main question is this solvable? I ...
1
vote
2answers
42 views

Solve x for a quadratic equation (not finding zeroes)

With a linear function $f(x)=5x+2=q$ can be solved for $x$ by rewriting it as $x=(q-2)/5$ While with a quadratic function $f(x)=5x^2+2x+2=q$ how would you solve for both x's on one side? So you ...
-2
votes
3answers
35 views

Find p and q for y(x)=x^2+px+q [on hold]

Find $p$ and $q$ for $y(x)=x^2+px+q$ if the function has minimum equal to $-4$ for $x=1$ Can anyone try to solve this please?