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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111
votes
16answers
10k 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
397 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 = ...
7
votes
3answers
256 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
209 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
116 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
359 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
3answers
298 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 ...
6
votes
4answers
127 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
304 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
93 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
275 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
4k 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
167 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
893 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
573 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
103 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
129 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
724 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
596 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
123 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
143 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
294 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
361 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
786 views

Where did $-4x$ come from?

I'm going over my quadratic equations for the ACT and I came across this quadratic: $$(x – 2)^2 – 12$$ My teacher said we could have factored it out into this: $$x^2 – 4x – 8$$ But I just don't ...
4
votes
5answers
386 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
425 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
76 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
89 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
148 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
128 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
64 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
72 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
178 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
92 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
577 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
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
2answers
177 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
107 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 ...
4
votes
2answers
190 views

Solving awkward quadratic equation to obtain “nice” solution.

I would like to solve the following quadratic equation to get a "nice" analytic solution for $\rho$. $\rho^2(r\sin\theta-2nr^2)+\rho(2nr^3-2r^2\sin\theta-2\sin\theta+2nr)-2nr^2+3r\sin\theta=0$ where ...
4
votes
1answer
626 views

Relationship Between Roots and Coefficients of a Quadratic

To prove this lemma I use the relationship between roots and coefficients of a quadratic equation but did not get the result. Please help me prove this lemma. If ‎‎ $ - ‎\theta‎‎_{2}x^2 - ‎ ...
4
votes
0answers
45 views

Computing question: A quadratic which gives primes [closed]

This is about Project Euler Problem 27. The question is: Considering quadratics of the form $n^2 + an + b$, where $\lvert a \rvert < 1000$ and $\lvert b \rvert < 1000$ Find the product ...
4
votes
2answers
584 views

Determine if equation will generate perfect squares

Given the following quadratic equations: $4n^2 + 128n - 131$ $4n^2 + 16n - 11$ $4n^2 + 24n - 3$ Is it possible to determine how many values of n will generate a perfect square? Or better yet, is ...
3
votes
4answers
240 views

Where is the order of the variables inside the parentheses coming from?

I'm reading Sawyer's Prelude to Mathematics, here: I can't understand what's the meaning and application of "condition" here. Also when he gives the example on the cubic equation, stating that ...
3
votes
5answers
228 views

Proving Quadratic Formula

purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become ...