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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121
votes
17answers
12k 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 ...
25
votes
4answers
4k 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?
19
votes
10answers
3k views

Why is every equation always equal to zero? [closed]

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 equal to zero? Why have mathematicians defined it ...
17
votes
3answers
403 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 = ...
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 ...
13
votes
0answers
142 views
+50

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 = ...
11
votes
8answers
604 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
315 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
0answers
21 views

Intersection of two hyperplanes

$G$ and $H$ are hyperplanes in $\mathbb{P}_n$ with coordinates $g=(g_0, \ldots, g_n)$, $h=(h_0, \ldots, h_n)$. How can I find a symmetric matrix $A_Q$ of a quadric $Q$ with $ Q = G \cap H$, where ...
7
votes
3answers
259 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
211 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
123 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!
7
votes
6answers
373 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 ...
6
votes
9answers
2k 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
152 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 ...
6
votes
1answer
305 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
1answer
85 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 ...
6
votes
2answers
98 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
288 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
5k 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
171 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
451 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
2answers
915 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
677 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
111 views

What is the minimum value of $abc$

If the roots of the equation $$ax^2-bx+c=0$$ lie in the interval $(0,1)$, find the minimum possible value of $abc$. Edit: I forgot to mention in the question that $a$, $b$, and $c$ are natural ...
5
votes
2answers
139 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
3answers
240 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 ...
5
votes
3answers
786 views

How to solve problems involving roots. $\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$

How to solve problems involving roots. If we square them they may go to fourth degree.There must be some technique to solve this. $$\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$$
5
votes
2answers
648 views

Find equation of quadratic when given tangents?

I know the equations of 4 lines which are tangents to a quadratic: $y=2x-10$ $y=x-4$ $y=-x-4$ $y=-2x-10$ If I know that all of these equations are tangents, how do I find the equation of the ...
5
votes
4answers
124 views

How do you solve $4x^2=-16x$? I get different answers depending on the method used.

I'm solving the following GRE problem: Solve $4x^2=-16x$ Method 1: I simply divide both sides by $4x$ :$$x=-4$$ Method 2: I solve by factoring:$$4x^2+16x=0$$ $$4x(x+4)=0$$ $$x=-4, x=0$$ Using ...
5
votes
2answers
167 views

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$. I made quite a few attempts but could not ...
5
votes
1answer
324 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 ...
5
votes
2answers
86 views

System of Pythagorean Quadratics

I have a system of quadratics, obtained from three mechanical links, fixed at one end and free at the other. The intersection point of the three free ends is required. ...
5
votes
2answers
443 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 ...
4
votes
5answers
392 views

Why are quadratics factored into 2 brackets?

Why is it that no one seems to factor quadratics into just one bracket Eg: $$2x^2+8x+6$$ into $$2x\left(x+4+3\cdot\frac{1}{x}\right)\quad\text{or}\quad 2x\left(x+4+\frac{3}{x}\right)\quad ?$$
4
votes
4answers
525 views

Quadratic function concepts

My teacher was explaining quadratics in my class and it was a little bit unclear to me. The problem was Suppose $at^2 + 5t + 4 > 0$, show that $a > 25/16$ . My teacher said that there are ...
4
votes
3answers
79 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 ...
4
votes
7answers
92 views

How to find $x^2 - x$?

I'm quite a novice when it comes to maths. I'm on a problem in which I have had to isolate $x$ , through factorials which I completed without problem. However, now I am stuck on a seemingly more minor ...
4
votes
3answers
155 views

Values of $a$ for which $(a+4)x^2-2ax+2a-6 <0$ for all $x \in R$

How can we find all values of $a$ for which the inequality $(a+4)x^2-2ax+2a-6 <0$ is satisfied for all $x \in R$? For the given condition, $D >0$, therefore $ (-2a)^2-4(2a-6)(a+4) >0$. ...
4
votes
2answers
130 views

Finding the Extrema of a Function (without differetiation)

$$ (t^2-t+1)/(t^2+t+1) $$ prove that the function is upper bounded by 3 and lower bounded by 1/3 without differentiation
4
votes
2answers
66 views

Evaluate $a+b+c+d$

If $a$, $b$, $c$, and $d$ are distinct integers such that $$(x-a)(x-b)(x-c)(x-d)=4$$ has an integral root $r$, what is the value of $a+b+c+d$ in terms of $r$? I tried to analyze graphically by ...
4
votes
2answers
73 views

Integral values of an expression

Let $b=\sqrt{a^2+5a+8}-\sqrt{a^2-3a+4}$ Find number of integral values of b. My $long$ way using Calculus : Find domain of function : $R$ Note that function is continuous Prove the function is ...
4
votes
2answers
217 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 ...
4
votes
3answers
97 views

Find the number of values of $a$?

Consider a quadratic equation; $$ x^2 + 7x – 14(a^2 + 1) = 0,$$ … (where $a$ is an integer) For how many different value of $a$, the equation will have at least one integer root? I found out its ...
4
votes
1answer
627 views

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ with $a, b, c\in \mathbb N$, prove that $\log_5 {abc}\geq2$. The equations I could form are: 1) $f(0)>0$ and ...
4
votes
2answers
184 views

Find all values of a for which the equation $x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0$ possesses at least two distinct negative roots

Find all values of a for which the equation $$x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0 $$ possesses at least two distinct negative roots. I am able to prove that all roots would be negative .How to proceed ...
4
votes
1answer
41 views

$(a - 1)x^2+3(a + 1)x+4(a - 1) = 0$ has real solutions iff $7a^2 - 50a + 7\leq 0 $

How can we show that $(a - 1)x^2+3(a + 1)x+4(a - 1) = 0$ has real solutions if and only if $7a^2 - 50a + 7\leq 0$? I know these are quadratics and can solve them, but I'm not entirely sure what the ...
4
votes
2answers
107 views

Getting a standard form quadratic from a set of points ($3$ points)

I came home from school today, pulled out my homework, now I'm stumped. I don't want the answer, I just want to know how to do it. Here is the question that I'm reading: Determine a quadratic ...
4
votes
1answer
110 views

Necessary and sufficient conditions that the difference of two quadratic equations has no solutions in $\mathbb{N}$

Suppose you have an equation of the form $$ a(n^2 - m^2) + b(n-m) + c = 0 $$ With given integers $a$, $b$ and $c$. Is there a necessary and sufficient condition that the equation has no solutions ...