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 $y=a(x+b)(x+c)$.

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20 views

How to bind conditions on a Quadratic/Quadratic function [duplicate]

For example,we need to find the range of a for which expression $\dfrac{ax^2+3x-4}{3x-4x^2+a}$ assumes all real values for real values of $x$ how to proceed?
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4answers
50 views

What is the equation for this circle? [closed]

Circle C is plotted on a graph. Circle C's center is (0,0). A tangent of circle C goes through the points (0,10) and (-30,0). What is the equation for the circle and how do you calculate this? Circle ...
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2answers
149 views

Solving a quadratic equation problem with two variables

This is a post of two three problems regarding the method to solve bivariate quadratic equations. In brief, How does the elimination happen here. Or, how is the elimination used? (Update, I know how ...
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2answers
41 views

Using the discriminant to find the value of k.

Question:- $x^2-4x-1=2k(x-5)$ has two equal roots. Calculate the possible values of $k$. I know that that must mean the discriminant must equal $0$. So I found: $b = (-2k-4)$ $a = 1$ $c = (10k-1).$ ...
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Finding the inverse function of $f(x) = -(9x^2 + 6x + 2)e^{-3x}$ using the generalised Lambert function.

I just learned about the generalized lambert function and I was trying to use it to find the inverse of a function. I have solved the equation below, so can someone check if it is correct? Thanks in ...
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1answer
136 views

How do you approach when completing the square?

If $M = 3x^2 - 8xy + 9y^2 - 4x + 6y + 13$, where $x,y\in\mathbb R$, then $M$ must be: a) positive $\qquad$b) negative $\qquad$c) $0 \qquad$ d) an integer I somehow managed to figure it out by ...
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1answer
80 views

Graph of quadratic $f(x)=ax^2+bx+c$ when $a$ is fixed and $b,c$ are varied

I noticed a small thing while playing with the graph of quadratic. $$ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 + c - a\left(\frac{b}{2a}\right)^2$$ Clearly $b,c$ only determine how the vertex of the ...
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3answers
87 views

Would an equation in the form ax + b = c/x be considered quadratic?

For example is $2x + 3 = \frac 5x$ quadratic? On the one hand it has two solutions, $x = 1$ and $x = -5/2$ which is the number of solutions we'd expect from the fundamental theorem of algebra but on ...
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3answers
97 views

Sum of squares of consecutive integers equals to a square [duplicate]

I am not at all mathematics guy, just had a question. How can I find possible pairs of consecutive integers whose sum of squares equals to a square? I understand equation will be something like: x² + (...
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85 views

Mistake in MIT paper? [closed]

This is MIT paper: https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/readings/MIT8_01F16_chapter14.9.pdf Look at equations $(14.8.26)$ and $(14.8.27)$. Did I forget how to solve ...
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1answer
52 views

Two Quadratic Equation having real roots

Let $x^2 + 2ax + b = 0$ and $x^2 + 2bx + a = 0$ have real roots $(a,b > 0)$, then minimum possible integral value of ab is___________ My approach is as follow $T(x)=x^2 + 2ax + b = 0$, hence $4a^2-...
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24 views

Lagrange's Resolvent [closed]

So I saw a method to solve cubic equations using Lagrange's Resolvent on brilliant.org . Can this method be used on all cubics? Why does no one ever use this method? Is it better than the Cardano ...
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1answer
26 views

Find interval of $c$ such that $2e^{2x} -(c+1)e^x +2 \ge 0$ for all $x\in R$

Now I know the normal method of manipulation which will get us $$c+1\le 2(e^x + \frac{1}{e^x})$$ ie. $c\le 3$ But can I do it by assume $e^x=t$ and then resolving the quadratic? What complications ...
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460 views

Elementary solutions of the equation of a quadratic formula

We know that when $A=0$, the quadratic equation $Ax^2+Bx+C=0$ has one solution, $-\frac{C}{B}$. Since for every quadratic equation, a quadratic equation has at most two solutions: $\frac{-B + \sqrt{B^...
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2answers
179 views

Number of roots of the equation $ax^2+ bx + c = 0$ in $(1,2)$

Let $a, b, c \in R, a \ne 0$ such that $a$ and $4a + 3b + 2c$ have the same sign. Then the number of roots of the equation $ax^2+ bx + c = 0$ that lie(s) in $(1,2)$ is(are)? I began by writing that $...
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2answers
95 views

When does a parabolic graph have integral x-intercepts and vertex? [closed]

Given a quadratic function $f(x) = ax^2+bx+c$ for integers $a$ and $b$ and $c$, what must be true of $a$ and $b$ and $c$ to guarantee the roots of $f$ and the location and value of the minimum/maximum ...
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0answers
28 views

Two quadratic equations having a common root

If quadratic equations $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$ have both their roots common then they satisy, $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$ But even if both ...
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28 views

Nature of roots of $f(f(x))=x$. [duplicate]

Let $f(x)$ be a quadratic function. Suppose $f(x)=x$ has no real root. Prove that $f(f(x))=x$ has no real root. My attempt: Let the complex numbers $z$ and $\bar z$ be the roots of $f(x)=x$. I was ...
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2answers
67 views

how to solve inequalities like $0.9<(1+x)^{1/2} - \frac x 2 <1.1$ [closed]

$0.9 < (1+x)^{1/2} - \frac x 2 <1.1$ The first part comes under the square root but the second part is just $x/2$ so I don't know how to go about finding the value of modulus of $x$ from this.
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Attempting to factor $(x^2)+5x=0$ via the grouping method.

Now, I am perfectly aware that I can easily factor this by taking out the common factor, so I would get $x(x+5)$. This way the roots of my equation would be $0$ and $-5$ What is confusing me is that, ...
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23 views

Relation between $|F(\sqrt{\alpha})^*/(F(\sqrt{\alpha})^*)^2|$ and $|F^*/(F^*)^2|$

I was wondering if there exists a relation between the number of non-zero square classes of a field $F$, $|F^*/(F^*)^2|$ and the number of square classes in a quadratic extension of said field $|F(\...
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1answer
26 views

design the equation of a decreasing parabola (between 2 points)

I am trying to come up with equations that can create a smooth downward decrease along $y$ so it falls from $0.997$ down to $0.990$. I am interested at the interval [N/5, N] along the $x$-axis. I ...
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3answers
77 views

Factoring the quadratic equation $3x^2-23x+14=0$

I'm having trouble understanding how to factor this equation. Let's go step by step: First I use the sum/product pattern: $$3x^2−2x−21x+14=0$$ Then I take the common factors: $$x(3x−2)−7(3x−2)=0$$ ...
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3answers
99 views

Why is this approach wrong?

If $\alpha$ and $\beta$ are the roots of $x^{2} - 4 x - 3$, then find $$\frac{1}{(\alpha-4)^{2}}+\frac{1}{(\beta-4)^{2}}$$ Solution This is the approach that gives wrong answer. \begin{array}{l} x^{2}...
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3answers
130 views

Why does Solving system of quadratic equations gives extra roots?

Consider these system of Equations \begin{align*} \begin{cases} x^2+4x+4=0\\\\ x^2+5x+6=0 \end{cases} \end{align*} For solving them We have Method 1- Subtract both equations So $-x-2=0$ Hence, $x=-2$ ...
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0answers
63 views

Quadratic Approximation Method to find the maximum of $f(x)$.

This is a quadratic approximation method to find the maximum of a function $f(x)$. The algorithm is as follows: Choose three points $x_0, x_1, x_2$ such that $f(x)$ is unimodal function on interval $[...
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2answers
57 views

Making sense of William Jones's solution of quadratic equations and notation

In his History of Mathematical notation, Cajori (1993) writes about Jones's approach to the solution of a quadratic equation as follows: William Jones, when discussing quadratic equations, says: &...
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1answer
73 views

Determine the largest interval $]\alpha,+\infty[$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{\sqrt{a^2+b^2}}<1$ for every $a,b \in ]\alpha,+\infty[$? [closed]

Let $a$ and $b$ be two (strictly) positive real numbers (i.e., $a,b \in ]0,+\infty[$). How to determine the largest interval $]\alpha,+\infty[$ (that is contained in $]0,+\infty[$) such that $$\frac{1}...
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4answers
92 views

Why do the vertices of $f(x) = ax^2 + bx + c$, when fixing $a$ and $c$ but varying $b$, lie on $g(x) = -ax^2 + c$?

As a bit of background context, I'm teaching a calculus course this semester, and for a learning activity, the students had to examine various functions and how certain modifications affected them. In ...
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1answer
44 views

Isolating x in a quadratic expression by completing the square

I came across some study notes regarding quadratic expressions and there is a solution that I am having a hard time understanding. In the study notes it is stated: "The variable x in the ...
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2answers
108 views

Factoring $\frac{n(n+1)}2x^2-x-2$ for $n\in\mathbb Z$

I was factoring quadratic polynomials for high-school practice and I noticed a pattern: $$\begin{align} x^2-x-2 &=(x+1)(x-2) \\ 3x^2-x-2 &=(x-1)(3x-2) \\ 6x^2-x-2 &= (2x+1)(3x-2) \\ 10x^2-...
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3answers
107 views

Replace square root by +-*/ [closed]

I have this quadratic equation $$ x = a \, (1 + y \, b) \, y \, c $$ in which $a$, $b$ and $c$ are known constants. Given $x$, I need to find $y$. However, since this is a quadratic equation, the ...
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1answer
40 views

A little bit different quadratic Gauss sum

Could anyone give me a direction on how to demonstrate that $$ \sum\limits_{k = 0}^{N-1} e^{-i\frac{\pi}{N}(k+C)^2} = \sqrt{N} e^{-i\frac{\pi}{4}}, $$ if $N\in \mathbb{N}$ is even, for any $C \in \...
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1answer
68 views

solve quadratic inequality for all cases

I am in great danger. I am solving a general quadratic inequality $ax^2+bx+c \ge 0$ for $a \neq 0$ as follows: Let $$ax^2 +bx+c = 0,$$ then the critical points are: $$x= \frac{-b \pm \sqrt{b^2-4ac}}{...
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2answers
148 views

A method to derive the quadratic formula

I wonder if this is a valid method to derive the quadratic formula. It is not an optimal method at all, but I am intrigued if it is a genuine way or has any gap. The method starts from the equation ...
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1answer
42 views

Roots of $\frac{{2{P^2}}}{x} + \frac{{3{Q^2}}}{{x - 1}} = 6$

Let $\alpha ,\beta $ are the roots of $\frac{{2{P^2}}}{x} + \frac{{3{Q^2}}}{{x - 1}} = 6,P,Q \in {R_0}$ where $R_0$ represent non-zero real number (A) $\alpha ,\beta \in \left( {0,1} \right)$ (B) $\...
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2answers
30 views

Prove the following for the given quadratic

If the difference of the roots of $x^2-px+q=0$ is unity then prove that $p^2-4q=1$ and $p^2 =4 q^2=({1+2q})^2$ What I Tried 1.I proved the first part of the question using the understanding of the ...
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1answer
55 views

Is there a method to factor equations with two variables raised to the second power?

I found the equation $2b^2-ab-a^2=0$ on a problem and couldn't find a way to factor it. Is there any method to factor these types of equations?
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0answers
114 views

Why is the graph of a quadratic $y=x^2$ a parabola? [duplicate]

I had this question and saw the other thread, and the proof of why all quadratics were parabolas started with the axiom that $y = x^2$ is a parabola. I don't really understand why $y = x^2$ is a ...
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6answers
164 views

How can I prove that all quadratic equations are not injective?

I was trying to prove that any quadratic formula ($ax^2 + bx + c$) will not be injective, but I have a little problem. I started by assuming $f(x) = f(y)$. We can put x and y into the general form of ...
5
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1answer
106 views

How can one solve this without expanding.

\begin{array}{l} \text { If } a+b+c=1, a b+b c+c a=2 \\ \text { and } a b c=3 \text {. What is the value } \\ \text { of } a^{4}+b^{4}+c^{4} \text { ? } \end{array} This can be solved by expanding but ...
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2answers
72 views

Wht cannot this term be equal to zero? [closed]

I came across the following problem: ...
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1answer
57 views

Find the value of given terms using the quadratic equation [closed]

Let $a$ and $b$ be the roots of $x^2-px-(p+c)$ where $c$ not equal to $1$ Then 1.Find the value of $\frac{a^2+2a+1}{a^2+2a+c} + \frac{b^2+2b+1}{b^2+2b+c}$ 2.If p q are the roots of $(x-a)(x-b)=c$ ...
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2answers
59 views

Find the quadratic equation from given relatioship

the quadratic equation whose roots are a and b where $a^2 +b^2=5$ and $3(a^5+b^5)=11(a^3+b^3)$ What I Tried $a^2 +b^2=5$ $(a+b)^2-2ab=5$ $(\text{sum of roots})^2 -2(\text{products of roots})=5$ $...
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1answer
30 views

sum of roots of quadratic equal to the reciprocals sum

If the sum of the roots of the quadratic equation $ax^2 +bx +c+0$ is equal to sum of the squares of the reciprocals then $b^2 /ac +bc/a^2$ is equal to? What I tried I equated the sum of the roots and ...
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3answers
37 views

Quadratics Distance-Speed-Time Rate Word Problem

I have the following problem from a Textbook A small plane is travelling between Windsor and Pelee Island (a distance of approx. 60 km) and is directly affected by the prevailing winds. Thus, the ...
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0answers
23 views

Distance/similarity measure between quadratic curves

I'm working on a problem in image processing where I am looking to group detected curves. Basically I'm looking for a good metric of the similarity between a pair of quadratic curves in a discrete ...
2
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4answers
101 views

Finding all $a$ such that $f(x)=\sin2x-8(a+1)\sin x+(4a^2+8a-14)x$ is increasing and has no critical points

Find the set of all values of the parameter $a$ for which the function, $$f(x)=\sin\left(2x\right)-8\left(a+1\right)\sin\left(x\right)+\left(4a^2+8a-14\right)x$$ increases for all $x\in\Bbb{R}$ and ...
2
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3answers
158 views

Weird root of an equation

I was doing this math and I found myself dealing with this equation: $$(5-y)^{2}=(1-y)^{2}$$ Now, square rooting both sides we get, $$5-y=1-y...(i)$$ or, $$5-y=-1+y...(ii)$$ Now, (ii) gives us a ...
1
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4answers
77 views

Why does the function $0.5x^2+0.5x$ have all integer inputs output the sum of all integers before it? [duplicate]

I was messing around with sums in desmos and tried to see if any of these sums had other functions that were similar to them. The one I was messing with was just summing up all the positive integers ...

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