Questions tagged [quadratics]

Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

257
votes
21answers
31k 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 ...
6
votes
4answers
749 views

Quadratic substitution question: applying substitution $p=x+\frac1x$ to $2x^4+x^3-6x^2+x+2=0$

By using the substitution $p=x+\frac{1}{x}$, show that the equation $$2x^4+x^3-6x^2+x+2=0$$ reduces to $2p^2+p-10=0$. I can't think of anything that produces a useful result, I tried writing p as $p=\...
1
vote
9answers
987 views

How to factor $9x^2-80x-9$? [closed]

How do I factor a trinomial like this? I'm having a lot of difficulty. How do I deal with the $9x^2$?
39
votes
11answers
9k views

Why can a quadratic equation have only 2 roots?

It is commonly known that $ax^2+bx+c=0$ have two solutions $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ but how to prove that another root couldn't exist? I think derivation of quadratic formula is not enough......
2
votes
3answers
342 views

How to get the correct angle of the ellipse after approximation

I need to get the correct angle of rotation of the ellipses. These ellipses are examples. I have a canonical coefficients of the equation of the five points. $$Ax ^ 2 + Bxy + Cy ^ 2 + Dx + Ey + F = 0$...
3
votes
2answers
364 views

A.P. terms in a Quadratic equation.

The terms $a,b,c$ of quadratic equation $ax^{2}+bx+c=0$ are in A.P. and positive. Let this equation have integral root $\alpha,\ \beta$. Then find the value of $\alpha+ \beta + \alpha \cdot \beta$ ? ...
5
votes
2answers
15k views

Solve the equation $x^2+\frac{9x^2}{(x+3)^2}=27$

Problem Statement:- Solve the equation $$x^2+\dfrac{9x^2}{(x+3)^2}=27$$ I have tried to turn it into a quadratic equation so as to be saved from solving a quartic equation, but have not been ...
3
votes
5answers
3k views

How to factor the quadratic polynomial $2x^2-5xy-y^2$?

How do I factor this polynomial: $2x^2-5xy-y^2$ ?
2
votes
2answers
152 views

How to prove that the roots of this equation are integers?

Let there be an equation $a^2 + 4ab + b^2 - 121 = 0$ where I want to prove that a,b are integers. Then I want to find whether there are integer values of $b$ for which $a$ is also an integer. Let us ...
2
votes
3answers
974 views

Proving the second root of a quadratic equation

If $\alpha$ is a root of the equation $4x^2+2x-1=0$, then prove that $4\alpha^3-3\alpha$ is the other root. How do I proceed? The sum of the roots, the product of the roots lead me nowhere. Should I ...
2
votes
1answer
283 views

Method to eliminate $x$ between the equation $x^2 + ax + b = 0$ and $xy+ l(x + y) + m = 0$

If by eliminating $x$ between the equation $x^2 + ax + b = 0$ and $xy+ l(x + y) + m = 0$, a quadratic in $y$ is formed whose roots are the same as those of the original quadratic in $x$. Then ...
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 ...
10
votes
1answer
165 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 ...
11
votes
3answers
6k views

Numerically stable algorithm for solving the quadratic equation when $a$ is very small or $0$

Solving $a x^2 + bx +c=0$ for $x$ gives $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \text{, for } a \ne 0$$ But for $a = 0$ we get $$x=-\frac{c}{b}$$ How to implement a numerically stable algorithm ...
1
vote
2answers
96 views

A polynomial with integer coefficients that attains the value $5$ at four distinct points

There is a polynomial $f$ of integer coefficients such that $\deg(f) \geq 4$. Let's assume that there are four integers $a,b,c,d$ for which $f(a)=f(b)=f(c)=f(d)=5$. Prove that there is no integer $k$ ...
5
votes
4answers
509 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
3
votes
2answers
825 views

LOGARITHMIC INEQUALITY - TWO HOURS

Source: Brilliant Solve the inequality $$ \ x \log_{\log_{|x^2 - 3 | - 2 } (x^2 - 3|x| + 2) } \left( \dfrac{x^3 - |3x+2|}{x^3 - |3x-2|}\right) \geq 0$$ I have tried this many times, but I keep ...
4
votes
3answers
155 views

Distinct roots of $ax^2-bx+c=0$ in $(0,1)$, where $a,b,c\in \mathbb{Z}^+$

Question Statement:- Let $a,b,c$ be positive integers and consider all the quadratic equations of the form $ax^2-bx+c=0$ which have two distinct real roots in $(0,1)$. Find the least positive ...
3
votes
2answers
1k 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 ...
11
votes
8answers
755 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 ...
8
votes
2answers
19k views

How to find the solution of a quadratic equation with complex coefficients?

I know how to find the solution for a quadratic equation with real coefficients. But if the coefficient changes to complex numbers then what is the change in the solution? Want an example of such ...
5
votes
3answers
301 views

Range of a Rational Function

How to find the Range of function $$f(x)= \frac{x^2-3x-4}{x^2 - 3x +4}$$ I tried to equate the expression to $y$, then cross multiplied $$ y= \frac{x^2-3x-4}{x^2 - 3x +4}$$ $$ y(x^2 - 3x +4)= x^2-3x-...
3
votes
5answers
389 views

How to “Re-write completing the square”: $x^2+x+1$

The exercise asks to "Re-write completing the square": $$x^2+x+1$$ The answer is: $$\left(x+\frac{1}{2}\right)^2+\frac{3}{4}$$ I don't even understand what it means with "Re-write completing the ...
0
votes
4answers
871 views

Proving a quadratic polynomial has no real roots without using derivatives or any formulas

How do I prove that a quadratic polynomial has no real root without the use of any derivatives or formulas? The specific equation is $$x^2-6x+10=0$$
5
votes
3answers
2k views

Convert quadratic bezier curve to parabola

A quadratic Bézier curve is a segment of a parabola. If the $3$ control points and the quadratic Bézier curve are known, how do you calculate the equation of the parabola (which is an $y=f(x)$ ...
2
votes
6answers
402 views

Factoring Quadratics

Is there a method to find which numbers to use when simplifying quadratics? For example $x^2 + 5x + 6$ is easy enough to find, but what if I have bigger numbers, or I have this quadratic expression: ...
2
votes
2answers
88 views

Problem related to quadratic equations

Let $f(x) = x^2 + ax + b$ be a quadratic polynomial in which $a$ and $b$ are integers. If for a given $n$ , $f(n)f(n+1) = f(m)$ for some integer $m$ then the value of $m$ is ? Do we have to ...
1
vote
3answers
3k views

Rotate the graph of a function?

How do I rotate a graph of a function around a point, and show it in the related equation? An example could be $f(x)=\lvert x\rvert$ (absolute Value) and $f(x)=x^2$
0
votes
5answers
163 views

How do you factor a quadratic expression, without using the formula?

I am asked to factor $2x^2 -3x+1=0 $ using factorization, but I run into fractions, and it becomes very messy and complicated to deal with, especially since specifically asked not to use the formula. ...
19
votes
3answers
483 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{b^...
28
votes
6answers
6k 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 true....
9
votes
1answer
517 views

What methods are known to visualize patterns in the set of real roots of quadratic equations?

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied to the set of real roots of quadratic equations $ax^...
2
votes
4answers
258 views

Prove: For odd integers $a$ and $b$, the equation $x^2 + 2 a x + 2 b = 0$ has no integer or rational roots.

If $a$ and $b$ are odd integers, prove that the equation $$x^2 + 2ax + 2b = 0$$ has no integer or rational roots.
12
votes
1answer
279 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+...
3
votes
1answer
164 views

Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$.

Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$. Restrict the values of $p$ in $[0,2\pi]$. The given equation has real roots if: $$\cos^2 ...
2
votes
2answers
2k views

Proving that $\cos\frac{2\pi}{13}+\cos\frac{6\pi}{13}+\cos\frac{8\pi}{13}=\frac{\sqrt{13}-1}{4}$

How can I prove that: $\cos\frac{2\pi}{13}+\cos\frac{6\pi}{13}+\cos\frac{8\pi}{13}=\frac{\sqrt{13}-1}{4}$ Without using complex numbers? I tried to raise by 2 and to multipy by 2, and got: $2y^2=3+...
10
votes
5answers
7k views

Determining if a quadratic polynomial is always positive

Is there a quick and systematic method to find out if a quadratic polynomial is always positive or may have positive and negative or always negative for all values of its variables? Say, for the ...
5
votes
1answer
4k 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 $f(1)&...
4
votes
7answers
454 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 $\dfrac{b^...
4
votes
3answers
48k views

Find $xyz$ given that $x + z + y = 5$, $x^2 + z^2 + y^2 = 21$, $x^3 + z^3 + y^3 = 80$

I was looking back in my junk, then I found this: $$x + z + y = 5$$ $$x^2 + z^2 + y^2 = 21$$ $$x^3 + z^3 + y^3 = 80$$ What is the value of $xyz$? A) $5$ B) $4$ C) $1$ D)...
3
votes
1answer
296 views

General method for determining if $Ax^2 + Bx + C$ is square

Is there a general method for solving Diophantine equations in the form $Ax^2 + Bx + C = k^2$, preferably turning them into Pell's equations, when possible? For example, $2x^2 + x + 1 = k^2$ or $5x^2 +...
0
votes
6answers
195 views

$\omega$ is a solution of $x^2+x+1=0$, find $\omega^{10}+\omega^5+3$ [duplicate]

I am working on a scholarship exam practice assuming high school or pre-university math knowledge. I am stuck at the question below: Let $\omega$ be a solution of the equation $x^2+x+1=0$. Then $\...
0
votes
3answers
4k views

max of $e$ with $a+b+c+d+e=8$ and $a^2+b^2+c^2+d^2+e^2=16$ [closed]

Given that a,b,c,d,e are real number such that: $\begin{cases} a+b+c+d+e=8\\ a^2+b^2+c^2+d^2+e^2=16 \end{cases}$ determine the maximun value of $e$. I started like that : $\max(e)=\max(8-a-b-c-d)=\...
6
votes
2answers
1k 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
5answers
940 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 ...
1
vote
1answer
193 views

Determine all the values of the parameter $a$ for which the inequality $3-|x-a|>x^2$ is satisfied by at least one negative $x$.

I wanted to know, how can I determine all the values of the parameter $a$ for which the inequality $3 - |x-a| > x^2$ is satisfied by at least one negative $x$. I tried for $x<a, |x-a|=-(x-a)$ ...
0
votes
1answer
213 views

Two circles intersection

Could you tell what are all the four points in following? Two circles intersect at two points maximum when we want to draw intersecting circles. But there we are solving quadratic equations, what is ...
5
votes
1answer
188 views

I have a inequality, I don't know where to start

Show that for $x,y,z > 0$ the inequality is true: $\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}+x+y+z \geq \frac{(x+y)^2}{y+z}+\frac{(y+z)^2}{z+x}+\frac{(z+x)^2}{x+y}$ I have tried Holder, but i had ...
4
votes
2answers
120 views

Intuition for why equations of the form $k^x=x^c$ are not solvable trivially?

It recently occurred to me that I did not know how to solve equations of the form $k^x=x^c$ for any two constants $k$ and $c$. After much pain in algebraically manipulating the equation (using ...
4
votes
1answer
53 views

find $f(\frac{1}{2014})+f(\frac{2}{2014})+…+f(\frac{2013}{2014})$ of $f(x)=\frac{2}{2+4^x}$

$f(x)=\frac{2}{2+4^x}$ find $f(\frac{1}{2014})+f(\frac{2}{2014})+.....+f(\frac{2013}{2014})$ Please guide me through it, the only step I know is probably to eliminate the denominator ps. Not a ...