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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2answers
26 views

State the coordinates of the vertex and the number of $x$-intercepts for the following function

State the coordinates of the vertex and the number of $x$-intercepts for the following function: $$ y = -4x^2 + 1 $$ I am not really asking for a straight-up answer. If you could please tell ...
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1answer
46 views

Solving quartic equation using substitution

We are learning a lot about the history of our famous mathematicians and this specific one is stumping me. They want us to solve a problem a specific way and I can't seem to figure out how to do it. ...
2
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0answers
73 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
0
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2answers
58 views

Question on proving quadratic inequality

Let $ax^2+bx+c$ = 0 be a quadratic equation and $\alpha$,$\beta$ are real roots. Condition for $\alpha < -1$ and $\beta > 1$. Show that $1 +\frac{c}{a}$ + $\left|\frac{b}{a}\right| < 0$. ...
2
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2answers
31 views

Quadratic Polynomial with complex coefficients

Let polynomial $p(z)=z^2+az+b$ be such that $a$and $b$ are complex numbers and $|p(z)|=1$ whenever $|z|=1$. Prove that $a=0$ and $b=0$. I could not make much progress. I let $z=e^{i\theta}$ and ...
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0answers
34 views

Let $a,b,c$ be the sides of triangle. No two of them are equal and $\lambda\in\Re$…

Problem : Let $a,b,c$ be the sides of triangle. No two of them are equal and $\lambda\in\Re$. If the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real, then find the range of ...
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2answers
44 views

$2x^2-16x+28$ into standard form

I think I'm just doing something stupid here, because I know it's not hard. Here's what I did: $$y-28+{\_\_\_}=2x^2-16+{\_\_\_}$$ $$y-28+{\_\_\_}=2(x^2-8+{\_\_\_})$$ $$y-28+16= 2(x^2-8+16)$$ ...
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4answers
63 views

Secondary solving method of polynomial

$$x+1+\frac{1}{x}=0$$ This is a fairly trivial and possibly bland equation to solve. But for the sake of the question I will display them here: $$x\left(x+1+\frac{1}{x}\right)=x(0)$$ $$x^2+x+1=0$$ ...
1
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3answers
48 views

Do perfect square trinomials only have one root?

I apologize for the basic question, but I'm just now learning of perfect square trinomials in my math class. Google hasn't provided any relevant answers. Throughout all of the examples I have been ...
2
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3answers
334 views

Finding the roots of a different Quadratic equation from the roots of a Given Quadratic equation

The Question: If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$... Then find the roots of the equation $ax^2-bx(x-1)+c(x-1)^2=0$ My Attempt: The new equation can be ...
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3answers
38 views

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

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?
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2answers
43 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 ...
1
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1answer
66 views

Quadratic polynomials describe the diagonal lines in the Ulam-Spiral

I'm trying to understand why is it possible to describe every diagonal line in the Ulam-Spiral with an quadratic polynomial $$2n\cdot(2n+b)+a = 4n^2 + 2nb +a$$ for $a, b \in \mathbb{N}$ and $n \in ...
1
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2answers
67 views

solve $\sqrt{x+7}<x$ for $x\in \mathbb{R}$

solve $\sqrt{x+7}<x$ I tried $\sqrt{x+7}<x\\ x+7<x^2\\ x^2-x-7>0\\ x\in \left(-\infty, \dfrac{1-\sqrt{29}}{2}\right) \cup \left( \dfrac{1+\sqrt{29}}{2},+\infty\right) $ I m not ...
3
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3answers
39 views

solve $|x-6|>|x^2-5x+9|$

solve $|x-6|>|x^2-5x+9|,\ \ x\in \mathbb{R}$ I have done $4$ cases. $1.)\ x-6>x^2-5x+9\ \ ,\implies x\in \emptyset \\ 2.)\ x-6<x^2-5x+9\ \ ,\implies x\in \mathbb{R} \\ 3.)\ ...
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2answers
19 views

graph quadratic form and find the equation of asymptotes

So I had this quadratic form that need to be graphed showing both original and new axes. And I also need to find out the equation of asymptotes. $$ \left\{ \begin{aligned} ...
0
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3answers
63 views

Solve $3^{2x} -2 \cdot 3^{x+5} + 3^{10} = 0$ for $x$

Here's the question: Solve for $x$ in $$3^{2x} - 2 \cdot 3^{x+5} + 3^{10} = 0$$ I know that you have to factor something out, I'm just not sure what that something is. Thanks in advance
3
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1answer
38 views

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ ...
2
votes
2answers
115 views

Epsilon-Delta proof of $\lim_{x\to 2} x^2=4$

I have seen and understand the delta-epsilon proof of the limit of $x^2$ for $x\to2$, such as explained here: https://www.youtube.com/watch?v=gLpQgWWXgMM Now I am wondering, is there also another ...
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1answer
24 views

Solving a quadratic equation with complex coefficient

Express $z4$=-$\sqrt{3}$+i in polar form. Hence solve the equation $Z^2$=$z4$ for $z$ a complex number. You may leave the answer in polar form. My answer: $z4$ in polar form is 2cis-30$^{\circ}$ and ...
2
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2answers
295 views

Proving the an expression is larger than a simplified quadratic

Let p and q be positive real numbers. Prove that $$ (p + 2)(q+2)(p+q) \ge 16pq $$ Any explanation/answer would be extremely helpful. Thanks : )
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4answers
621 views

Why are the coefficients always equal?

Take the equation $ax^{2} + bx + c = 3x^{2} + 4x + 53$. Why is it always true that $a = 3, b = 4$ and $c = 53$? I've seen many examples like this where the coefficients are equated, and was just ...
1
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1answer
30 views

Why does this hyperboloid change into a surface? [duplicate]

Given this equation $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z=6$ and the corresponding quadric: If I rearrange the equation to $(x+y+z-3)(x+y+z+2)=0$ (which is equivalent), I get: So, which is the right ...
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3answers
51 views

When do variables cancel out?

Sometimes if I randomly combine different equation and try to solve for a variable, one of them will cancel out. Why? For example: $\displaystyle x^2 = 4y^2$ and $\displaystyle x = 2y + 1$ And solve ...
1
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2answers
45 views

What type of equation is this? How to solve it?

$$m^4+a^4=0$$ , the answer I obtained is $$0+i1,0-i1$$ but the answer is given as a/sqrt(2)-a/sqrt(2),a/sqrt(2)+a/sqrt(2)
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1answer
39 views

solving the equation by reducing it to quadratic form

Solve for $x$ $$4^x-\dfrac{3^x}{\sqrt3}=3^x\cdot\sqrt3-\dfrac{2^{2x}}2$$ I don't understand how to convert it into quadratic equation how should I equate all bases
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2answers
52 views

Alternative Quadratic Formula

Well the formula for solving a Quadratic equation is : $$\text{If }\space ax^2+bx+c=0$$ then $$x=\dfrac{-b \pm \sqrt{b^2 -4ac} }{2a}$$ But looking at this : [Wolfram Mathworld] (And also in other ...
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votes
1answer
100 views

Using a number system other than the decimal [duplicate]

Travelling in the Kingdom of Crystal Skull, Indiana Jones discovered a small box with notation $$ 3 x^2 - 25 x + 66 = 0 \implies x_1 = 4,\; x_2 = 9, $$ which seems to be incorrect. However after ...
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2answers
40 views

Quadratic equations / Perfect square

I am dealing with a programming problem and was wondering if there's a general method to solve it. Is there a general way to find 'n' such that: $a^2n^2 + bn - c$ is a perfect square? a, b, c given ...
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0answers
22 views

Find Next Position and Velocity from Instantaneous Values

To find the position of an object at a given point in time: $y_0 + v_0t - \frac{32t^2}{2} = y_t$. And to find the object's speed at a given point in time: $v_0 - 32t = v_t$ So say I give the ...
1
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2answers
53 views

Why $ax^2+bx+c = a(x-r)(x-s)$, where $r$, $s$ are the roots?

When I was reading about math, I came across the following - Suppose the roots of the quadratic $ax^2+bx+c$ are $r$ and $s$. Then $ax^2+bx+c = a(x-r)(x-s)$ for all values of $x$. Is there ...
2
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2answers
98 views

My brother asked me to explain a algebra problem. How should I explain it?

So the problem is: $$\frac{4}{x}+\frac{6}{2}=x$$ And we solved it using the pq formula. But than he asked me: How do I know when I should apply pq to similar equations like this and not just: ...
5
votes
3answers
635 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 ...
0
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4answers
45 views

Find $f(a)$, $f(a + h)$, and the difference quotient, given $f(x)=7-8x+2x^2$

Question is here: I need help with part c. I tried plugging everything in and simplifying to a point where my final answer was $(-8h+2ah+2h^2)/h$ My work: $(7-8a-8h+2(a+h)^2-7+8a-2a^2)/h$ ...
1
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0answers
37 views

Differential Equations: Confocal Ellipse and Hyperbola

I am currently brushing up on Conic Sections, and I am having some problems on solving a first order quadratic differential equation. I would appreciate any help on the topic! I know that confocal ...
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5answers
76 views

Could translate/explain this for me?

I have this problem: $$ 10x^2 - 7x - 12 = 0 $$ And apparently the method to factoring it is to find two numbers whose product is the same as the product of the coefficient of $x^2$ and the constant ...
3
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1answer
34 views

Find the number of equations having real roots.

If both $a$ and $b$ belong to the set $\{1,2,3,4\}$ , then number of equations of the form $ ax^2+bx+1=0$ having real roots is $a.)\ 10\\ \color{green}{b.)\ 7}\\ c.)\ 6\\ d.)\ 12\\ $ To solve ...
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1answer
39 views

not easily factored quadratic expression how to find its roots [closed]

Could you please help me and explain this issue: If a quadratic equation is not easily factored then its roots can be found using quadratic formula: If $ax^2+bx+c=0$ ($a\ne0$), then the roots are ...
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5answers
90 views

Sum of real solutions on equation $\sqrt{\sin^2{x} + {1 \over 2}} + \sqrt{\cos^2{x} + {1 \over 2}} = 2$ in interval $[0,2\pi]$ is?

I know that solution is $4\pi$ but I do not know how do they get to this solution. I always get that $x \in R$ and that $-1 < \cos 2x < 1$ when converting it to double angle. EDIT : So ok, I ...
3
votes
1answer
50 views

$ a + a^2 = 90 $. Need hint.

From trig text. Am supposed to find complementary angles $a$ and $a^2$. Tried completing the square, got $(a + 1/2)^2 = 361/4$. Stuck. Help.
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2answers
43 views

Quadratic question problem

I have been asked to look at a yr 12 question about a space ships trajectory modelled by a quadratic equation. But my solution to the question has a negative minimum for distance at a time of T = 13 ...
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5answers
60 views

How to find value of $c$ in $ax^2+bx+c=0$ having the $a$ and $b$?

Problem: solve (find roots): $ 7x^2-3x=0$. How to find $c$ in order to solve using the Quadratic Formula?
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7answers
67 views

How to factor $ax^{2}+bxy+cy^{2},\,a\neq 0$?

Question: Factor: $3x^{2}-5xy-12y^{2}$ Answer: $(x-3y)(3x+4y)$ What are the exact steps to finding this answer from the original question (factored form from standard form, respectively)?
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2answers
23 views

Higher Order Polynomial Function Solver

I have a 5th order, uni-variable, polynomial :( As I understand the only way to solve this is to guess? Since this is a real world equation, rather than something from a textbook, there really isn't ...
1
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2answers
34 views

Proof for showing $1 + \frac{c}{a} + \frac{b}{a} > 0$ for any quadratic equation,where its roots are non real.

The question is as follows : If the equation $ax^2 + bx + c$ has non real roots, prove that $1 + c/a + b/a > 0 $. Looking at the question,the first thing that came to my mind was to use ...
0
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2answers
66 views

finding maxima and minima of a quadratic equation

I'm dealing with a quadratic equation(with 2 independent variable) which looks like: $$f(x,y) = 15.390x^2 - 0.001y^2 - 0.003xy - 69.985x + 0.263y + 58.740 $$ But I'm not being able to determine the ...
1
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1answer
22 views

Quadratic equation with greatest integer functioin

$[x]^2-7[x]+12=0$ find $x$? where $[x]$ is Greatest Integer function I have tried to solve the question like this: putting $[x]=y$ , we have the equation: $y^2-7x+12=0$ by solving this equation ...
0
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3answers
47 views

Nature of roots of a quadratic equation with irrational co-efficients

What would be the nature of the roots of the equation $$2x^2 - 2\sqrt{6} x + 3 = 0$$ My book says that as the discriminant is 0 so the roots are rational and equal. But discriminant can be used for ...
9
votes
5answers
556 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?
0
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3answers
47 views

Complete The Square for $x^2-10x+18$

Okay so the question is: $x^2-10x+18$ has to be written in the form $(x-a)^2+b$ and I have to provide the values of $a$ and $b$. I worked out that $a= -5$ and $b= -7$. On the video I am watching, ...