Questions on quadratic equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-h)^2+k$ or $y=a(bx+c)(cx+d)$.

learn more… | top users | synonyms

0
votes
2answers
15 views

Let f be a continuous function defined on [-2009,2009] such that f(x) is irrational for each $x \in [-2009,2009]$ …

Problem : Let f be a continuous function defined on [-2009,2009] such that f(x) is irrational for each $x \in [-2009,2009]$ and $f(0) =2+\sqrt{3}+\sqrt{5}$ Prove that the equation $f(2009)x^2 +2f(0)x ...
0
votes
1answer
37 views

Quadratics and roots

The question I am trying to solve is this: $4 x^2 - 3 x - 3 = 0$ has roots $p, q$. Find all quadratic equations with roots $p^3$ and $q^3$. I was able to answer this question by simply finding the ...
0
votes
3answers
39 views

Find the set of real numbers ($x$ not equal to zero) such that $2x + 1/x < 3$.

Pretty straightforward question, I just had a question for the conclusion. I rearranged, and factored and have the quadratic: $$2x + 1/x < 3$$ (multiply both sides by x and rearrange) $$2x^x - ...
1
vote
2answers
50 views

Proving a simple equation with complex numbers

Fix $A \in ℂ$ and $B \in ℝ$ Let $z \in ℂ$. Show that the equation $|z^2| + Re(Az) + B = 0$ has solutions iff $|A^2| ≥ 4B$ I have no trouble proving the forward implication, but its the "only if" ...
0
votes
0answers
15 views

How to minimize this quadratic function?

As described at page 3 of this document, I need to minimize the following quadratic function: $E(w,x,y,z) = \sum_i \frac{(w-T_i(x,y,z))^2}{1+|\Delta f(x_i,y_i,z_i)|^2} $ where $w=f(x,y,z)$ and ...
5
votes
2answers
144 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 ...
1
vote
1answer
41 views

Irreducible quadratic factors; partial fraction decomposition.

Please help me understand why there is Dx+E, Fx+G etc, instead of the regular A's, B's, C's etc. What is it about the irreducible quadratic in the denominator that makes it different on top?
1
vote
2answers
59 views

Solve $f(x) = ax^2 + bx + c$ to find the value of $K$

$f(x)=ax^2+bx+c$, where $a=-9$, $b=12$ and $c=16$. If $$-1<f'(x)<1$$ then $h<x<k$. To $2$ decimal places, what is the value of $k$? Hi, this is working for solving $f(x) = ax^2 + bx + ...
0
votes
0answers
14 views

Quadratics and function question

A quadratic function is given by ${h(x) = ax^2 + bx + c}$ where ${a}$, ${b}$, and ${c}$ are all nonzero real numbers. The function ${h(x)}$ intersects the x-axis at two distinct points and satsifies ...
0
votes
1answer
23 views

Quadratic roots question

If $3.5 - {\sqrt 2}$ and $3.5 + {\sqrt 2}$ are the roots of a quadratic equation ${ax^2 + bx + c = 0}$; then which of the following is not correct? A. a is nonzero - I ruled out this because if was 0 ...
3
votes
0answers
67 views

Fibonacci Quadratic Residue

After some research I have came up with a conjecture on fibonacci quadtratic residue: ...
0
votes
0answers
41 views

About a Variant of Ulam Spiral

Here I read about a variant on the Ulam spiral: [A] structure may be seen when composite numbers are also included in the Ulam spiral. [...] Using the size of the dot representing an integer ...
2
votes
1answer
38 views

What is the greek symbol that represents the ratio of the length and with in a rectangle.

I understand that there is pi, but I was provoked by a question that asked this, "I have a rectangle and if I cut a square off of it it produces the same type of rectangle. I know that the width is 1 ...
0
votes
0answers
30 views

Calculating $\arg\min_x (1-\Phi(x;\mu_1,\sigma_1^2)+\Phi(x;\mu_2,\sigma_2^2))$

I would like to find $x$ satisfying the following expression: $$\arg \min_x R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2)$$ where $$R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) ...
0
votes
0answers
17 views

Ellipse equation. What does it need to be in order for $b > a$?

We have the quadratic equation: $$ax^2 + bx + cy^2 + dy + e$$ $a$ and $c$ are both negative or both positive. How can I, by looking at that only, determine whether $b$ (the length of the semi-minor ...
2
votes
4answers
52 views

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take?

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take? This is what I have done: Let $y = 3^x$ $$9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$$ $$\implies9y^2 + (t^2 - 4t - 2)y + ...
0
votes
2answers
52 views

How factor with square root

I have the following equation that I'm trying to factor, but I'm stuck at the end. $$\frac{zx^{-4}\sqrt{x}(yz^4)^3}{z^7xy}$$ $$\frac{\frac{1}{x^4}\sqrt{x}(yz^4)^3}{z^6xy}$$ ...
1
vote
3answers
26 views

Need help with basic factoring equation

I'm just trying to brush up on my factoring of quadratic equations. $$\frac1{x+3} + \frac1{x^2 + 5x +6}$$ $$\frac1{x+3} + \frac1{(x+2)(x+3)}$$ $$\frac{(x+2)(x+3) + (x+3)}{(x+2)(x+2)(x+3)}$$ Then ...
3
votes
1answer
76 views

Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
2
votes
0answers
107 views

Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real root…

Problem : Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real roots belonging to the interval $(1,2) $ then the minimum possible values of a is ...
-1
votes
2answers
27 views

Quadratic equation form?

Suppose we know that the sum of two positive numbers is $2k$ and their product is $m$ then which of the following will be its quadratic equation and why? 1) $x^2$+ $(2k)x$+ $m$= $0$ 2) $x^2$- ...
0
votes
4answers
69 views

When do you use the quadratic formula?

I am revising for a mathematics exam and am looking over simultaneous equations. I was curious as to when I use the quadratic formula and when I don't? I realize there are multiple ways to solve a ...
1
vote
1answer
53 views

how to solve these two quadratic equations

Can someone help me find the solution for these two quadratic equations ? $ 2(z^2) \ - \ 3.023bz \ + \ 0.115(b^2) \ + \ 2.0814b \ + \ 0.142z \ - \ 0.5856 \ = \ 0 $ $ 6.0828(z^2) \ + \ 2.0414bz \ + \ ...
0
votes
1answer
20 views

What is the minimum possible $ non $ integral value of a

Let a A subscript(m) (m=1,2,3,....p) be the possible integral values of a for which the graphs of $ f(x)=ax^2+2bx+b $ and $g(x)=5x^2-3bx-a$ meets at some point for all real values of b. 1) What is ...
3
votes
1answer
170 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
2
votes
1answer
23 views

A question on multinomial theorem using binomial theorem

$(3x^2+2x+c)^{12}=\sum A_r x^r$ and $\frac{A_{19}}{A_5}=\frac 1 {2^7}$ Find $c$. I really have no idea what to do with this. This was on a test. I have studied only binomial theorem. So, please ...
0
votes
1answer
46 views

Determining quadratic function of this word problem

I have this word problem in my homework: ...
1
vote
1answer
69 views

A Cubic Equation

$2x^3+ax^2+bx+4=0$, $(a,b \in R^+)$ has three real roots. Then : A. $a\geqslant 4.2^{\frac 1 3}$ B. $a\geqslant 1.2^{\frac 1 3}$ C. $a\geqslant 6.2^{\frac 1 3}$ D. $a\geqslant 2.2^{\frac 1 3}$ ...
1
vote
1answer
26 views

Terminology of “linear”, “quadratic”, etc. for multi-input functions

It is my understanding that, according to typical math terminology: The function $f(x, y) = x + y$ is "linear". Specifically, it's linear in both $x$ and $y$, but this is understood implicitly. ...
1
vote
2answers
108 views

number of integral values for which $x^2+19x+92$ is a perfect square.

number of integral values of x for which $x^2+19x+92$ is a perfect square=? I have no idea how to do this. Please help.
1
vote
0answers
21 views

Variable quadratic functions

I have some doubts in proving the following:- C is the curve $$y = \frac 1 {k+1} [2x^2 + (k + 7)x + 4]$$, where k is a real number not equal to -1. Show that C always passes through two fixed points ...
0
votes
2answers
132 views

How would I solve the quadratic $x^2+3x-70=0$?

How would I solve the following quadratic equation $$x^2+3x-70=0 $$ This is my attempt below $$(x-7x) (x+10x)=0 $$ $$ x-7x=0 \implies -6x=0 \implies x=6$$ $$x+10x=0 \implies 11x=0 \implies ...
0
votes
1answer
59 views

How to interpret coefficients in polynomial regression?

I am working on my thesis (study) about poverty incidence rate and its socio-economic factors using second-order polynomial regression without interaction. The final model in my study is ...
0
votes
1answer
21 views

Verification of solutions to some polynomial prob/

$\boxed{\text{Problem 1}}$ Find the other solution of the equation $(1+\sqrt3)x^2-(5-\sqrt3)x+6-6\sqrt3=0$ given that $2$ is a solution Ma solution: $x_1\cdot x_2=\tfrac ca$ therefore letting ...
3
votes
2answers
37 views

prove for p(x) which is a quadratic polynomial

$p(x)$ is a quadratic polynomial . Prove that any given number for $a$ with one exception , we can find a number $b$ such that $p(a)=p(b)$ and $a$ is not equal to $b$.
0
votes
2answers
90 views

Solving for $a$ and $b$

How would you solve the equality: $$a\,b\,\left(b+a\right)=1$$ in terms of a and/or b? Would you subtract 1 from both sides and work from there? Or would you simply expand and work from there?
0
votes
2answers
38 views

How to use $t(29/\sqrt{2})<0$ where $t(x)=x^2-41x+420$ to prove that $41/29<\sqrt{2}<42/29$??

So I was investigating different ways to approximate $\sqrt{2}$. Here's my latest: $$Let:t(x)=x^2-41x+420$$ then the roots of $t(x)$ are $20$ and $21$. I showed that then $t(x)=(x-20)(x-21)$ and ...
4
votes
2answers
73 views

Integral values of an expression

Let $b=\sqrt{a^2+5a+8}-\sqrt{a^2-3a+4}$ Find number of integral values of b. My $long$ way using Calculus : Find domain of function : $R$ Note that function is continuous Prove the function is ...
0
votes
2answers
34 views

What does each term in the trinomial represent and their relation to each other? [duplicate]

I am trying to get a grasp or concept understanding what how trinomials answer questions other than answering questions in an algebra class. I. Looking for the practical application. What does the ...
0
votes
1answer
32 views

With Trinomial's, can someone explain the purpose of the first, second and third term. In layman terms.

I know that the first term is a quadratic and I suppose that lets us know we are dealing with identifying a curve, and the third term is our constant. I just can't quite put it all together as how ...
2
votes
2answers
46 views

Find the range of values of $k$ for which the following equation has real roots.

Find the range of values of $k$ for which the following equation $x^2+(1-k)x=k$ has real roots. I tried it, for real roots, $b^2-4ac \geqslant 0$ $x^2+(1-k)x-k=0$ $(1-k)^2-4*1*(-k)\geqslant0$ ...
19
votes
10answers
3k views

Why is every equation always equal to zero? [closed]

A linear equation is $$ ax + b = 0 ; \,\, \,\, a\neq 0 $$ A quadratic equation is $$ax^2 + bx + c = 0 ; \,\, a\neq 0 $$ And so on... Why are all equal to zero? Why have mathematicians defined it ...
0
votes
2answers
88 views

Solving quadratics using $y=a(x-p)^2+q$ ?

The vertex is $(-4, 2)$ and the y-int is $(-3,-1)$. How would I solve this, or find out what "$a$" is so I can write the equation and graph it?
1
vote
3answers
110 views

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$.

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$. First of all, I solved $1-x<(x-1)(5-x)<3$ which gives me $(x-1)(x-6)<0$ and $(x-4)(x-2)<0$. How to find the range, ...
1
vote
1answer
54 views

Found an example for solving via quadratic formula in a book where I am wondering if this is correct

As a refresher, I was skimming through a free Calculus online textbook "MOOCULUS massive open online calculus" (https://mooculus.osu.edu/handouts) and stumbled upon the following example solving a ...
-2
votes
2answers
43 views

Find the range of values of $x$ for the inequality $x^2-4x-1>0$ [closed]

Find the range of values of $x$ for the inequality given. $x^2-4x-1>0$
1
vote
5answers
33 views

Linear systems from positions a quadratic passes through?

I don't understand the question. $$y = ax^2 + bx + c$$ passes through the points $(1,-4),(-1,0),(2,3)$. Write down a linear system (unknowns $a,b,c$) of three equations relating the unknowns to each ...
0
votes
5answers
112 views

Show that $3x^2-4x+2$ is always greater than $0$.

How do I show that the function $3x^2-4x+2$ is always greater than $0$?
1
vote
1answer
331 views

Find the maximum or minimum value of the quadratic function by completing the square.

Find the maximum or minimum function of the quadratic function by completing the squares. State the value of $x$ at which the function is maximum or minimum. $y=3x^2+7x+9$ I already posted similar ...
3
votes
2answers
2k views

Find the maximum or minimum value of the quadratic function.

Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of $x$ at which the function is maximum or minimum. $y=2x^2-4x+7$ $x^2$ has a coefficient ...