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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3
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2answers
53 views

Difference of the roots of quadratic formula

I have a question to solve with roots quadratic formula that is , $$a^3 + b^3 = (a+b)(a^2-ab+b^2)$$ $$(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$ but I didn't understand how the below formula is ...
9
votes
1answer
100 views

Replacing numbers by roots of quadratic

We have $10$ numbers in the interval $(0,1)$, not necessarily distinct. At any moment, we can choose two of them, $a$ and $b$. If the quadratic $x^2-ax+b$ has two (possibly identical) real roots, we ...
4
votes
4answers
166 views

Solutions of $z^6 + 1 = 0$

Solve: $$z^6 + 1 = 0$$ That lie in the top region of the plane. We know that: $$(z^2 + 1)(z^4 - z^2 + 1) = 0$$ $$z = -i, i$$ We need to solve: $$((z^2)^2 - (z)^2 + 1) = 0$$ $$z = \frac{1 \pm ...
0
votes
2answers
44 views

Intersections of two parabolas given focii

As part of Voronoi's algorithm, I need to calculate the intersection of two parabolas to compute a breakpoint at run time. I've spent literally 8 hours on this, and I've only gotten my equations to ...
1
vote
3answers
103 views

A quadratic polynomial is nonnegative for all $x$ if and only if the discriminant is nonpositive

Show that if $a>0$ the inequality $ax^2+2bx+c\ge 0 $ for all values of $x$ if and only if $b^2-ac\le 0$. I tried to prove it by: $ax^2+2bx+c≥ b^2-ac$. Used partial derivatives with respect to ...
2
votes
1answer
86 views

Is the simplest form of a quadratic equation factored form or standard form?

I've done a bit of research about what defines simplest form, but I could not find a clear answer. Suppose we had to choose: $$(x - 4)(x + 2) \quad\text{or}\quad x^2 - 2x - 8$$ A question asked me, ...
3
votes
5answers
793 views

Quadratic formula not working?

Suppose I want to find the $ n $ for which $$(n)(n+1)/2 = 10 \Longrightarrow n^2 + n - 20 = 0$$ Clearly a solution is $4$. But, suppose we wanted to find that solution by use of quadratic formula. ...
1
vote
3answers
41 views

Quadratic formula in double inequalities

I have the double inequality: $-x^2 + x(2n+1) - 2n \leq u < -x^2 + x(2n-1)$ and I am trying to get it into the form $x \leq \text{ anything } < x+1$ Or at least solve for x as the ...
1
vote
5answers
82 views

Why sometimes we get only one root of quadratic equations?

What is logic behind getting (sometimes) only one root of a quadratic equation which satisfies the equation?
1
vote
2answers
59 views

Avoiding extraneous solutions

When solving quadratic equations like $\sqrt{x+1} + \sqrt{x-1} = \sqrt{2x + 1}$ we are told to solve naively, for example we would get $x \in \{\frac{-\sqrt{5}}{2},\frac{\sqrt{5}}{2}\}$, even though ...
0
votes
3answers
54 views

find x again in equation

I asked a similar question but I wanted to be sure understand. I have to find $x$ in the equation $$x^2=-2x-1$$ I go to left and get $$x^2+2x+1$$ Then I use a similar trick used in similar question ...
4
votes
2answers
100 views

When the quadratic formula has square root of zero, how to proceed?

Is there an easier way to solve the following equation? $$x^2=2x-1$$ I think I know how to find $x$, using the quadratic formula: I get $$x^2-2x+1=0$$ then $$x=\frac{2 \pm \sqrt{4-4})}2= ...
1
vote
1answer
51 views

Quadratic Equations with the variable raised to a power higher than 2.

This is the problem in question - Solve the equation $ a^2 - 8a + 12 = 0 $. Hence find the four values of $x$ which satisfy the equation $ (x^2 - x)^2 - 8(x^2 - x) + 12 = 0 $. P.S. - I got the ...
1
vote
1answer
187 views

Symmetric Properties of Roots (Quadratic Roots)

What is the proof that - $α^2$ + $β^2$ = $(α+β)^2$ - 2αβ $α^3$ + $β^3$ = $(α+β)^3$ - 3αβ(α+β) $α^4+β^4$ = ($α^3+β^3$)(α+β) - αβ($α^2+β^2$) (α+β)4 = α4 + 6α3β + $4α^2β^2$ + $6αβ^3$ + $β^4$ ...
1
vote
2answers
28 views

Factoring Quadratic

I have used the substitution P = dy/dx to solve a first-order D.E of degree 4, so I got this: I have to show that the above statement can be written as: I tried to factor out first by taking p a ...
1
vote
0answers
48 views

Show Equivalence of Binary Quadratic Forms

I've been stuck on these two problems from my problem set for quite a while. Any help would be appreciated! 2)Suppose that $ax^2 + bxy + xy^2$ is equivalent to $Ax^2 + Bxy + Cy^2$. Show that $gcd ...
3
votes
3answers
50 views

How to rearrange a quadratic into its factorized form?

Like the title says, I'm a bit confused about how the smart people of past centuries figured out that the quadratic: $$ ax^2+bx+c = a(x-x_1)(x-x_2). $$ The book I have at hand shows how to do it the ...
4
votes
2answers
449 views

Solving an equation with exponentials

$$2^x+4^x+12=0$$ How exactly am I supposed to solve this? Am I supposed to get $x$ alone or solve it another way?
2
votes
1answer
42 views

Find cosine of acute angles in a right triangle.

If sides of a right triangle are in Geometric Progression, then find the cosines of acute angles of the triangle. [Answer] $\frac{\sqrt{5}-1}{2}$,$\sqrt\frac{\sqrt{5}-1}{2}$ My work: Using ...
0
votes
3answers
51 views

How do you reverse $\frac{100n(n+1)}{2}=c$ to find n given c?

I'm developing a game where the character experience needed by level is given by Gauss' formula multiplied by 100: $ \dfrac{100\mathrm{level}(\mathrm {level}+1)}{2}$. So the experience table is ...
2
votes
2answers
39 views

how to find A in quadratic projectile motion

what would the standard form be for this question? During a drumline performance, a drummer throws his drumstick with an upward velocity of 32 feet per second. if the drummer releases and catches the ...
1
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3answers
34 views

imaginary algebraic inequality equation

This problem was actually given to me as a typo. I decided to work it despite it being a typo and it presented a couple of questions regarding applying imaginary results to an inequality equation. ...
1
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2answers
36 views

Find the value of $2p+4q+7r$ given that $2p,\ q,\ 2r$ are in geometric progression.

It is given that $2p,\ q, \ 2r$ are in G.P. Also the roots of the quadratic equation $$px^2+qx+r=0$$ are of the form $\alpha ^2,\ 4\alpha -4$. Find the value of $2p+4q+7r$. From the given data: ...
1
vote
0answers
52 views

Real world application of slanted conics (parabolae especially)

I am writing a report on slanted conics of the form $$(x-h)^2+(y-k)^2= \dfrac{d}{\sqrt h}$$ Where $(h, k)$ is the focus, and $d$ is the directrix. Are there any real world applications for slanted ...
1
vote
1answer
18 views

If $a_0,a_1,a_2 \cdots a_{99} \in R$ and $f(x) =x^{100}+a_{99}x^{99}+a_{98}x^{98} +\cdots +a_0$ be such that $|f(0)|=f(1)$..

Problem : If $a_0,a_1,a_2 \cdots a_{99} \in R$ and $f(x) =x^{100}+a_{99}x^{99}+a_{98}x^{98} +\cdots +a_0$ be such that $|f(0)|=f(1)$ and each root of f(x) =0 is real and between 0 to 1. If product ...
0
votes
0answers
51 views

How to find the Coefficient of the Quadratic Term?

Given $4x^3 +bx^2+cx+d$ and two roots of this cubic function $(0,0)$ and $(2,0)$ Find the coefficient of the quadratic term? When I first read this I had no idea how to solve this and still ...
1
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0answers
25 views

All solutions to Quadratic matrix polynomials

I am after two things: 1- algorithms for finding all solutions of possibly large quadratic matrix equations of the form $AX^2+BX+C=0$ 2- (if possible) software implementing the algorithms ...
0
votes
0answers
74 views

How can I check these equations if they have a solution?

I have two equations which are: $p^3+k\equiv0 (mod \quad h) $ and $(3p^2+3mp+m^2)m\equiv 0(mod \quad h)$ where $k,h,m >0$ and $p\ge0$ and $h\nmid m$ I need to show for given k,m,h and for all ...
1
vote
1answer
107 views

Vertex Equation of an inverse quadratic function.

I'm working on a graphing web tool using JSXGraph, The user should be able to draw different functions. I was able to allow the user to draw quadratic functions by creating the vertex of the function ...
0
votes
2answers
20 views

What is the process of expanding quadratic equation

I am currently doing a math problem: $(a-b)(a^2+ab+b^2)$ However, I am not sure how I can actually expand this problem Do I multiply $(a-b)$ with each individual item within the other bracket?
2
votes
2answers
72 views

Quadratic equations: Why does factoring by grouping work?

We are learning factoring by grouping - The teacher explained the process but didn't explain the logic behind it. You need to multiply the coefficient on the x-squared term by the constant to get a ...
1
vote
5answers
48 views

How to find the equation for a parabola for which you are given two points and the vertex?

I was originally given the value $(4,-2)$ as the vertex of a parabola and told that it also includes the value $(3,-5)$. From this point, I deduced that the next point would have the same y-value as ...
0
votes
1answer
37 views

How to solve the equation $y = \frac{x^2}{20000} + 0.0046x + 62.054$ for $x$?

So I have an equation I am trying to solve for x. $y = \frac{x^2}{20000} + 0.0046x + 62.054$ I can solve it up until this part, and then my mind just blanks. $$\left(\frac{y-62.054}{0.0046}\right) ...
1
vote
3answers
71 views

Is there any methods to solve for integer solution of a quadratic equation like $ax^2 + bx + c = 0$

Is there any method to solve for integer solution of a quadratic equation like following: $$ax^2 + bx + c = 0$$ where $a, b, c \in \mathbb{Z}$ If not is it possible for the Special case: ? $$x^2 -x ...
0
votes
1answer
45 views

Quadratic equation formula for a,b,c from 3 points

I can solve for a, b, c given three points for a parabola for example (1,1)(2,4)(3,9) but i need to create a program which returns a,b,c in the form: ...
1
vote
1answer
41 views

Law of Indices and Quadratic Expressions

So I think I need some clarification about the rules for manipulating indices, in particular these two equivalences: $(x^3)^2 = x^{(3)(2)} = x^6$ $a = a^1$ Take the expression: $(5+5)^2$, which is ...
2
votes
1answer
31 views

What is the solution for this quadratic program?

Given scalars $p_1\geq p_2\geq \cdots \geq p_r > 0$, can we find a solution for following problem? \begin{align} \text{minimize} & & & \sum_{j=1}^{r} p_j (1-t_j)^2 \\ \text{s.t.} \\ ...
0
votes
2answers
39 views

If $ax^2+bx+6=0$ doesn't have $2$ distinct real roots, then find the least value of $ (3a+b)$

If $ax^2+bx+6=0$ doesn't have $2$ distinct real roots, then find the least value of $(3a+b)$ $a,b\in \mathbb R$ Any hint for this question?
0
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2answers
36 views

Quadratic Equations GRE Quants

It would be very useful if someone can give me an answer to this question with a proper explanation. One of the factors of the equation $x^2 +9x + c$ is $(x+11)$, where $c$ is a constant. Which of ...
0
votes
2answers
23 views

How to form a quadratic equation with real coefficients if $x_1=4-7i$?

Why is the quadratic equation $x^2-8x+65=0$? I tried to find $p$ and $q$ to form the equation but i need $x_2$ because: $$p=-(x1+x2)$$ $$q=x_1*x_2$$ so $x2=$?
2
votes
4answers
718 views

What does 'express in terms of $x$' mean?

For the following question : $f(x) = 2x^2 + 4x $ It asks me to express the following in terms of $x$: $f(-2x)$ What does the question mean by this? Does it mean make $x$ the subject?
0
votes
2answers
33 views

quadratic equation max and min problem

A transit company charges $1.25$ dollars per ride and currently averages $10,000$ riders per day. The company needs to increase revenue but found that for each $0.10$ dollars increase in fare the ...
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1answer
21 views

max and minimum qudratic function problem

A piece of wire $20$ metres long is cut into $2$ pieces and each piece is bent to form a square. Determine the length of the two pieces so that the sum of the areas of the two squares is a minimum. ...
1
vote
5answers
55 views

solve this equation for $x$ : $y=x-6\sqrt{x}$

solve for $x$ this equation : $$y=x-6\sqrt{x}$$ I've tried raising everything to the power of two but it doesn't work $x$ shouldn't have two values.
17
votes
2answers
313 views

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 = ...
0
votes
2answers
36 views

Factorize this polynomial $ax^2+bx+c$ into factors of the first exponent in the cases when D>0, D=0

The previous request was to prove the identity $ax^2+bx+c=a[(x+(b/2a)^2-(D/4a^2)]$, where $D=b^2-4ac$ And I proved it from the left to the right, which means I managed to express $ax^2+bx+c$ as ...
0
votes
1answer
18 views

Finding the y-vertex of a function and X2.

I am trying to solve the following exercise: The graph of the fuction $y=-2x^2+bx+c$ passes through the point (1,0) and has as its vertex the point (3,S). What is the value of s? Options: A -5_____ ...
0
votes
1answer
79 views

Discriminant of Quadratic with circle

The circle $x^2 + (y - c)^2 = r^2$, where $c > 0$ and $r > 0$, lies inside the parabola $y = x^2$. The circle touches the parabola at exactly two points located symmetrically on opposite sides ...
1
vote
0answers
17 views

Probabilty of even number of games won/lost uses auxiliary variables for a quadratic equation. Why?

In a problem of finding the probability that an even number of games (even S) not being lost in $l$ games, I read the following explanation : "We form the equation, $x^2 - 4rx + 2r^2 = 0$, and ...
2
votes
3answers
69 views

Quadratic Equations - One rational solution?

I have a question that I am working on: Which of the following will give one rational solution? 4x^2 = 9 4x^2 - 12x = -9 x^2 = 5 x^2 - 2x + 14 = 0 2x^2 = x I am ...