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

learn more… | top users | synonyms (1)

6
votes
3answers
80 views

Finding the values of $q$ for which the quadratic equation $qx^2-4qx+5-q=0$ will have no real roots.

Find the values of $q$ for which the quadratic equation $qx^2-4qx+5-q=0$ will have no real roots. So I've gotten as far as using the discriminant to find the values of $q$, but I'm stuck on the last ...
2
votes
4answers
80 views

Why does this method to solve a quadratic equation for $x$ omit $x=0$?

Here is a simple quadratic equation: $$9x^2 - 36x = 0$$ We proceed as following: \begin{align*} 9x^2 & = 36x\\ 9x & = 36\\ x & = 4 \end{align*} So, we get $x=4$. But, here's another ...
7
votes
1answer
179 views

Geometric derivation of the quadratic equation

The quadratic equation can be thought of as specifying distances in the Euclidean plane. It tells us that the $x$-intercepts of a function occur at a distance of $\frac{\sqrt{b^2-4ac}}{2a}$ from the $...
3
votes
1answer
35 views

Why are the factors of some solutions to a Pell equation also a solution?

I came across this observation while trying to answer this post using the Pell equation $x^2-2y^2=1$. Define, $$P(m) = \frac{ (3+2\sqrt{2})^m+(3-2\sqrt{2})^m}{2}$$ $$Q(m) = \frac{ (3+2\sqrt{2})^m-(3-...
1
vote
9answers
307 views

Why is $x=2 \implies (x-2)(x-3)=0$ false?

Let $P(x)$ be the equation $x=2$ and $Q(x)$ be the equation $(x-2)(x-3)=0$ By definition of implication I see that $P(x)$ implies $Q(x)$... As I see it, any premise that is false can give any ...
2
votes
2answers
96 views

Derive quadratic formula [duplicate]

I cannot understand how quadratic formula to solve for $x$ was derived. On this website, it explains the steps Following I understand but I cannot understand how they got $b/2a$ and why they ...
12
votes
1answer
87 views

Aside from the obvious stuff, do the partial functions that solve the quadratic equation have any interesting properties?

Let us define partial functions $$f_+,f_- : \mathbb{R} \leftarrow \mathbb{R} \times \mathbb{R} \times \mathbb{R}$$ so as to return the zeros of the quadratic $ax^2+bx+c$ whenever they exist, such ...
3
votes
2answers
56 views

Complex number in quadratic equation

Find $a,b$ given that a root of $x= 1+2i$ and the equation $ x^2+(a+bi)x+2i-1=0$ I tried finding it by $\Delta$, which I got $\Delta=a^2+2abi-b^2-8i+4$ I tried substituting the root into the equation ...
0
votes
2answers
30 views

Given Function, find domain and description of graph $y = f(x)$

I am studying for Graduate Record Exam. The following question is difficult. Given the domain and description of $f(x) = 5 - (x + 20)^2$, including its shape, and the $x$ and $y$-intercepts To find ...
0
votes
1answer
40 views

Counting the solutions of a quadratic equation

I have read that a non-singular conic will contain $p+1$ points on the finite field $\mathbb{F}_{p}$, but there is a exercise on Silverman's Rational Points on Elliptic Curves, p.142, 4.8 that tells ...
0
votes
2answers
29 views

Bound on Coefficients

For real $a,b,c$ the following holds $|ax^2+bx+c|\le 1 ; \forall x\in [0,1]$.Show that $|a|+|b|+|c|\le 17$. Cant show that the equality holds.I always get the lesser bounds.
1
vote
1answer
83 views

A sequence of quadratic polynomials

Q. Let $p_n(x)=a_n x^2+b_n x$ be a sequence of quadratic polynomials where $a_n,b_n \in \Bbb R$ $\forall n \ge 1$. Let $\lambda_0$,$\lambda_1$ be distinct nonzero real numbers such that $\lim_{n \to \...
3
votes
2answers
136 views

System of quadratic equations over field of size 2

I am working on system of quadratic equations. \begin{cases} (\alpha_1^1x_1+\ldots+\alpha_n^1x_n)(\beta_1^1y_1+\ldots+\beta_m^1y_m)=0\\ \ldots \\ (\alpha_1^kx_1+\ldots+\alpha_n^kx_n)(\beta_1^ky_1+\...
0
votes
2answers
33 views

Graphing Parabolas Word Problem

A flying cannonball’s height is described by formula $y=−16t^2+300t$. Find the highest point of its trajectory. In how many seconds after the shot will cannonball be at the highest point? What is the ...
0
votes
1answer
29 views

Parabolas Inequalaties

If a vertical parabola opens upward, has its vertex in the third quadrant, and $y=ax^2+bx+c$ is the equation of this parabola, which of the following can be true? Sketch a curve for each possible case....
2
votes
1answer
26 views

Is there a term in mathematics for Metcalfe's Law?

Metcalfe's Law states that the value of a network is proportionate to the square of the number of users. This comes from the idea that there are $N*(N-1)/2$ pairs in a network of size $N$. Does this ...
1
vote
1answer
77 views

If $\alpha$ , $\beta$ are roots of the quadratic equation $x^2 -2p(x-4) -15 = 0$ , then answer the following .

What is the set of of values of $p$ for which one root is less than $1$ and the other is greater than $2$ ? A) $ (7/3,\infty) $ B) $ (-\infty,7/3) $ C) $ x \in R $ D) $ None $ Please tell me ...
3
votes
2answers
37 views

If $ ax^2 + 2bx + c = 0 $ and $ a_1x^2 + 2b_1x + c_1 = 0 $ have a common root , then prove the following. [closed]

If $a/a_1 , b/b_1 , c/c_1 $ are in A.P. then $ a_1 , b_1 , c_1 $ are in G.P. I have no idea , how to approach this . What I have thought : For the AP series $ a/a_1 = k - d $ the rest be k &...
1
vote
2answers
44 views

If $ 3x^2 + 2\alpha xy + 2y^2 + 2ax - 4y + 1 $ can be resolved into two linear factors, then prove the following.

Prove that : $ \alpha $ is a root of the equation $ x^2 + 4ax + 2a^2 + 6 = 0 $. What does it mean by "can be resolved into two linear factors"? If it means $( ax + b ) ( cx + d )$ , is it necessary ...
9
votes
2answers
133 views

$f(g(h(x)))=0$ has $8$ real roots

Find all quadratic polynomials $f(x),g(x)$ and $h(x)$ such that the polynomial $f(g(h(x)))=0$ has roots $1,2,3,4,5,6,7$ and $8$. I don't know what to do. Making a $8$ degree equation is quite tedious....
-1
votes
1answer
26 views

Deriving Quadratic Function Using Table of Values

How do you derive a quadratic function given its table of values when there is no zero x value given? For example, x|1|2|3 y|-6|-3|4 I tried two methods in solving this but I can only get a. Any ...
1
vote
2answers
38 views

Nature of roots of $x^2+2(a-1)x+(a-5)=0$

A quadratic equation is given as $x^2+2(a-1)x+(a-5)=0$ then what could be the possible value of a if: a) The equation has positive roots b) The equation has roots of opposite sign c)...
3
votes
1answer
50 views

Parabola conic section

Two tangents to the parabola $y^2= 8x$ meet the tangent at its vertex in the points $P$ and $Q$. If $|PQ| = 4$, prove that the locus of the point of the intersection of the two tangents is $y^2 = 8 (...
1
vote
1answer
30 views

Rewrite the equation $(x-a)^2 + (y-b)^2 = r^2$ to make $y$ a function of $x$

I'm trying hard to figure out how $(x-a)^2 + (y-b)^2 = r^2$ can be written as $y = b + \sqrt{r^2 - (x-a)^2}$. My book says that you’ll want to have $y$ as a function of $x$.
0
votes
1answer
41 views

Solving a system (3) of nonlinear equations

I want to solve the following euqations: $6y-2xz = 0 $ $6x-2yz = 0 $ $x^2 + y^2 -8 = 0 $ One way would be to multiply the first equations with y and the second with x and then substract the first ...
2
votes
0answers
38 views

How to use piecewise quadratic interpolation?

I'm attempting to get the hang of quadratic interpolation, in MatLab specifically, and I'm having trouble approaching the process of actually creating the spline equations. For example, I have 9 ...
4
votes
1answer
64 views

Change the variables in $Q(x,y,z)=(x-y+z-1)^2-2z+4$ to have $Q(f(u,v,w))=u^2+v$

I have a problem with this exercise. Initially, they gave me this polynom, and I had to complete the squares: $$Q(x,y,z)=x^2-2xy+2xz+y^2-2yz+z^2-2x+2y-4z+5.$$ I've done it, and I've checked with ...
3
votes
2answers
36 views

Condition for inverse of quadratic function - alternative solutions

I was helping my friend teacher to assemble a list of exercises to their precalculus students. So I came up with this problem: Let $f$ be a quadratic function, i.e. $$f(x) = ax^2 + bx + c,$$ ...
-3
votes
2answers
56 views

Exponential simultaneous equations, short multiplication formulas - only for sneaky people

I am obligated to do some exercises from some Russian maths book and I solved them, but the teacher told us to use the smartest possible way to achieve this and I guess mine aren't sneaky enough and ...
0
votes
1answer
28 views

About quadric classification by completing square

I'm doing a seminar of geometry. We're learning how to classify quadrics with Maple, and there's a steps we have to follow in order to find what kind of quadric we have. Initially, they give me this: ...
0
votes
1answer
36 views

Number of real solutions of quadratic equation

I have the following question that puzzles me: How do I determine the number of non-trivial real solutions to the general equation $ax^2 + bxy + cy^2 = 0$ (up to a scalar)? My attempt was to fix $y \...
1
vote
3answers
55 views

How to solve a problem with a variable in both the base and exponent on opposite sides of an equation

I am working on systems of equations in Pre-Calculus, and I presented the teacher a question that I had been wondering for a while. $x^2 = 2^x$ The teacher couldn't figure it out after playing with ...
0
votes
4answers
35 views

How do you distribute this negative?

So I have $-(x - 2)^2$. Do I rewrite it as $-(x - 2)\cdot-(x - 2)$ and distribute the negative to the inside making it $(-x + 2)(-x + 2)$ or add the negative at the end of doing FOIL?
1
vote
3answers
45 views

What is the difference between the two real numbers that satisfy this equation?

What is the absolute difference between the two real numbers $x$ for which $(x+1)(x-1)(x-2) = (x+2)(x+3)(x-3)$? Express your answer in simplest radical form I tried guessing solutions but seeing how ...
0
votes
2answers
51 views

Question on quadratic polynomials with real cofficients

Let$ P(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients. Suppose there are real number $s ≠ t$ such that $P(s) = t$ and $P(t) = s$. Prove that $ b - st$ is a root of the equation $ ...
1
vote
4answers
61 views

How do you factor $x^2-x-1$?

I know you can't have all integers, but how do you factor this anyway? Wolfram|Alpha gives me $-\frac{1}{4} (1+\sqrt{5}-2 x) (-1+\sqrt{5}+2 x)$. Cymath gives me $(x-\frac{1+\sqrt{5}}{2})(x-\frac{1-\...
0
votes
1answer
12 views

Determining Parabola by Equation Problem

I need some help figuring out how to solve this problem. Which of the following could be a graph of the equation $ y= ax^2 + bx + c$ where $b^2 - 4ac = 0$ The picture below was the correct answer. ...
0
votes
1answer
18 views

Let $P(x)=(m^2+4m+5)x^2-4x+7,m\in R$.If $3\leq x\leq 5$,then find the minimum of the minimum value of $P(x).$

Let $P(x)=(m^2+4m+5)x^2-4x+7,m\in R$.If $3\leq x\leq 5$,then find the minimum of the minimum value of $P(x).$ The minimum value of $P(x)=(m^2+4m+5)x^2-4x+7$ occurs at $x=\frac{2}{m^2+4m+5}$ So the ...
1
vote
2answers
68 views

Sum of all values of $b$ if the difference between the largest and smallest values of the function $f(x)=x^2-2bx+1$ in the segment $[0,1]$ is $4$

Find sum of all possible values of the parameter $b$ if the difference between the largest and smallest values of the function $f(x)=x^2-2bx+1$ in the segment $[0,1]$ is $4$. I found that the ...
2
votes
1answer
48 views

How many solutions to $x^2-4x-\cos x=-8$?

$$x^2-4x-\cos x=-8$$ We want the number of solutions. Now I tried taking $\cos x$ as constant but by formula for root we get a trig equation which can't be solved. Any help? Thanks!
-1
votes
4answers
92 views

If a quadratic polynomial $ax^2+bx+c$ has the form $(rx+s)^2$. Does it implies that $b^2-4ac=0$? [closed]

If a quadratic polynomial with integral coefficients $ax^2+bx+c$ is of the form $(rx+s)^2$ Can we say that the discriminant $$b^2-4ac=0?$$ If so how do you prove it?
1
vote
0answers
15 views

Linear Algebra and Quadratic Equations

I'm just wondering if Linear Algebra is concerned only with Linear equations? Can quadratic equations(or any higher power) also be considered under Linear Algebra? What does the term Linear stand for?
0
votes
1answer
16 views

I need help with variable expression

Good day! I have this coirdinate equation: $$\frac{gt^2}{2}+{v_y}t-\frac{5}{3}R=0$$ $$v=\sqrt{\frac{10}{9}*gR}$$ How i can express variable $t$ from this equation? I calculated this as quadratic ...
2
votes
0answers
57 views

Help solving the quadratic equation $ax^2-4bx+4bc-\frac{d^2}{a}=0$

I have been struggling to solve this quadratic equation in the variable $x$ with integral coefficients: $$ax^2-4bx+4bc-\frac{d^2}{a}=0$$ $a\neq 0$ of course.How do I ensure that $x$ is an integer? ...
0
votes
0answers
54 views

How many cows will eat the field?

Three pasture fields have areas of $\frac{10}{3}$, $10$ and $24$ acres, respectively. The fields initially are covered with grass of the same thickness and new grass grows on each at the same rate per ...
2
votes
2answers
41 views

If $a,b$ are the roots of the equation $x^2-2x+3=0$ obtain the equation whose roots are $a^3-3a^2+5a-2$, $b^3-b^2+b+5$

I have been trying this using sum of roots and product of roots but it gets too lengthy. So I found the roots of the given equation which are imaginary and tried to replace the values in the two given ...
1
vote
2answers
71 views

Prove Newtons method converges quadratically

f(x)= cosh(x) +cos(x) -3 Let x* be the none negative root of f. Prove that Newton's Method applied to f converges quadratically to x*. Really confused where to start for a proof. I understand that ...
1
vote
1answer
29 views

The number of integers $n$ such that the quadratic equation $nx^2+(n+1)x+(n+2)=0$ has rational roots is

The number of integers $n$ such that the quadratic equation $nx^2+(n+1)x+(n+2)=0$ has rational roots is $(A)0\hspace{1cm}(B)1\hspace{1cm}(C)2\hspace{1cm}(D)3$ The condition for the rational roots is ...
1
vote
0answers
23 views

solving quadratic equation in GF(2^m)

I am trying to implement Elliptic Curve Cryptography on software in GF(2^m). To do this, I need to be able to solve a quadratic equation, namely $x^2 + x = c$. After a lot of research, I know the ...