Questions on quadratic equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-h)^2+k$ or $y=a(bx+c)(cx+d)$.
1
vote
4answers
46 views
Quadratic Formula problem?
There is a right triangle. The hypotenuse is 17 units. The sum of the other two sides is 23. Find the length of the two other sides.
Thanks for everyone's help in advance!
1
vote
2answers
29 views
Maximum of d(12-d)
I'm a little confused on a quite simple quadratic problem. I need to calculate the maximum of $d(12-d)$ using basic quadratics. The answer is $6$ as can also be shown by $f'(x)= -2d +12$, however this ...
2
votes
1answer
71 views
Curve through four points — simple algebra??
The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms.
Suppose I want to ...
3
votes
3answers
75 views
Proving the quadratic formula (for dummies) [duplicate]
I have looked at this question, and also at this one, but I don't understand how the quadratic formula can change from $ax^2+bx+c=0$ to $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$. I am not particularly good ...
2
votes
1answer
51 views
Finding a prime $p$ to solve a quadratic congruence $\pmod{p}$
I have a congruence of the form $$ax^2+bx \equiv -1 \pmod{p},$$
where $p$ is an odd prime and $a,b \in \mathbb{Z}$. Given $a$ and $b$, is there a general method to finding $p$ such that the above ...
0
votes
1answer
50 views
Finding a polynomial of degree $n$ when value of $f(k)$ is equal to some value
Problem :
If $f(x)$ is a polynomial of degree $n$ and if $f(k) = \frac{k}{k+1}$ where $k =0,1,2,\ldots,n$, find $f(x)$.
Can we go like this : Let the polynomial be ...
3
votes
4answers
131 views
Maths GCSE very hard question
My friends recently took a Maths GCSE. In the paper, they came across a very difficult question which we spent a full half-hour train journey trying to figure out. We didn't manage it, so I've come ...
1
vote
1answer
25 views
Satisfying a condition on given quadratic equation
Let $P(x) = x^2 +2bx + c$ be a quadratic form where $b,c$ are real numbers.If $b^2 < c$ , show that $P(x) > 0$ for all $x$ .Is the converse also true?
The value of $x$ after solving the ...
0
votes
2answers
32 views
Quadratic and geometric average
I'd like to find the find the quadratic average and the geometric average. To do this I have these informations :
The standart deviation, the arithmetic average and the number of values.
I know the ...
1
vote
1answer
48 views
what if geometric sequence + geometric sequence
I wrote a program that basicly can find the formula of the sequence that created with any-degree equation.
For example if you give my program that sequence:
[1926, 2811, 833240, 28778265, 398155842, ...
1
vote
0answers
25 views
Quadratic Equation Modulo an even composite
I am familiar with using the quadratic formula and Tonelli-Shanks with Hensel's Lifting Lemma to solve a quadratic equation, but how do I solve a quadratic equation in an even modulus? I can't use the ...
1
vote
6answers
301 views
Is a Quadratic equation a function?
The definition of a function is "A function is a relation in which there is never more then one value of the dependent variable for every value of the independent variable."
Since a quadratic ...
1
vote
3answers
60 views
Solving quadratic equations by completing the square.
Graphing $y=ax^2+ bx + c$ by completing the square
Add and subtract the square of half the coefficent of $x$.
Group the perfect square trinomial.
Write the trinomial as a square of a ...
2
votes
2answers
70 views
sum of squares of the roots of equation
The equation is $$x^2-7[x]+5=0.$$
Here $[x]$ the greatest integer less than or equal to $x$. Some other method other than brute forcing. I tried a method of putting $[x]=q$ and $x=q+r$ which gives an ...
1
vote
1answer
31 views
Application of quadratic functions to measurement and graphing
thanks for any help!
Q1. Find the equation of the surface area function of a cylindrical grain silo. The input variable is the radius (r). (the equation is to be graphed using a graphics calculator ...
0
votes
0answers
62 views
If $f(y)=ax^2+bx+c$, does this imply that $x=\frac{-b \pm \sqrt{b^2-4a[c-f(y)]}}{2a}$?
The equation $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ is known as the quadratic formula and is the solution to the quadratic equation $ax^2+bx+c=0$.
Sometimes I encounter equations such as $x=y^2-y$. Is ...
0
votes
2answers
44 views
Irreducibility of quadratic polynomial in Z[x]
I would like to ask, how to test irreducibility of quadratic polynomial.
I found, that when square root of discriminant is integer, $\sqrt{D}\in Z, D=b^2-4ac$, the polynomial can
reduced. The document ...
1
vote
2answers
44 views
help with my hw its a quadratic equation [closed]
Write the quadratic equation using the following factor
$(7x-3) (-2x+1)$
$(R+9) (R-9)$
Factor: $x^2+6x+9$
Please help the hw is due tommorow
0
votes
2answers
129 views
Quadratic equation with tricky conditions. Need to prove resulting inequalities.
The roots of the quadratic equation $ax^ 2-bx+c=0,$ $a>0$, both lie within the interval $[2,\frac{12}{5}]$. Prove that:
(a) $a \leq b \leq c <a+b$.
(b) ...
1
vote
3answers
62 views
Why is the coefficient of $x$ in $\frac{1}{x}=0$?
I usually solve a quadratic equation:
$$ax^2+bx+c=0$$
Through a method I learned in school: For a monic quadratic, you make $x=y-\frac{b}{2}$.
The method is intended for a monic equation but in ...
5
votes
3answers
135 views
If $a+b=x$ and $ab=y$, what is the quickest way to solve for $a$ and $b$?
The mechanistic approach would be to simply substitute $b=y/a$ in the first equation to obtain a quadratic in $a$. But seeing the simplicity of the givens, I feel that there must be some better and ...
-1
votes
1answer
33 views
Quadratic Baseball Question
The height of a baseball is modeled by the function $h(x)=-0.005x^2+0.3x+1.5$, would an outfielder which is modeled by the function $m(x)=-0.06x+5.6$ where $50 \le x \le 90$, catch the ball?
-1
votes
3answers
64 views
Find the min value of $3a+b$
If $ax^2+bx+c=0$ has no real roots then find min value of $3a+b$ for $c=6$;
Please tell me how to proceed , i don't have any clue of what to do.
4
votes
9answers
390 views
Prove $ax^2+bx+c=0$ has no rational roots if $a,b,c$ are odd
If $a,b,c$ are odd, how can we prove that $ax^2+bx+c=0$ has no rational roots?
I was unable to proceed beyond this: Roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
and rational numbers are of the form ...
4
votes
1answer
77 views
Necessary and sufficient conditions that the difference of two quadratic equations has no solutions in $\mathbb{N}$
Suppose you have an equation of the form
$$
a(n^2 - m^2) + b(n-m) + c = 0
$$
With given integers $a$, $b$ and $c$.
Is there a necessary and sufficient condition that the equation has no solutions ...
0
votes
1answer
18 views
Quadratic Equation - Nature of roots
What is the product of real roots of the equation $t^2x^2+|x|+q=0$
Since the complex equation is positive so sum of the roots are positive, here I am having four option as answers :
$>0$
...
2
votes
1answer
47 views
Convexity of Quadratic equation Inequality?
Solving an inequality of the form $x^TAx\geq0$ or $x^TAx\leq0$ is straightforward. I mean we have to check if A is positive semidefinite or negative semidefinite. But what would be the solution to the ...
3
votes
1answer
36 views
Question on quadratic problem set
Okay so I have a quadratic function problem. I will omit the problem for now just because we don't really need it. My problem is: M is surface area. Do I have to write M(x, y) or just M in the area ...
0
votes
1answer
43 views
Is this quadratic word problem correct so far?
I'm a little confused as to how to solve this word problem I have. The problem is: A rectangular box (with a top) has a square base. The sum of the lengths of its edges is 8 feet. What dimensions ...
1
vote
1answer
53 views
How to show $\frac{300}{v} - \frac{300}{v+20} = 1.25$
A man travels a distance of $300$ km. On his return journey his average speed increased by $20$ km/h and his journey time decreased by $1\frac{1}{4}$ hours. If $v$ is the average speed of his outward ...
-2
votes
3answers
88 views
Determine the of p and other roots.
One of the roots of $3x^2 + p =5x$, is $2$. Determine the value of $p$ and the other root.
1
vote
2answers
57 views
Linear Regression to quadratic function
What is the optimal linear regression (w and w/o y-intercept) for a quadratic curve w.r.t. mean square error.
Mathematically speaking:
Given,
$$y = x^2$$
for
$$x = [-a,a]$$.
What is the best ...
2
votes
2answers
59 views
Help in understanding quadratic equation
Sorry if this is a complete dummy question, but I haven't done math in years and I'm quite rusty. I'm reading this explanation of least squares regression, which internally uses the quadratic equation ...
0
votes
2answers
42 views
is there an analytic solution to $n^2+kn-d=m^2$ m,n integers
For $k=24,d=-17;m=8,n=3$, completing the square gives $(12+n)^2=m^2+161$ Where $161$ just happens to be the product of two primes $(q=7,p=23)$, so for large $k,m,n$ factoring may be very slow.
...
2
votes
4answers
146 views
Simplifying $\{ x \in \mathbb{R} : x^2 - x - 6 \geq 0 \}$ when possible
Simplify the following interval notation when possible:
$$\{ x \in \mathbb{R} : x^2 - x - 6 \geq 0 \}$$
1
vote
1answer
66 views
Find value of $k$
For what value of $k$, are the roots of the quadratic equation
$$(k+4)x^2 + (k+1)x +1 = 0$$ equal.
0
votes
2answers
82 views
jenny farm and the dozen egg ???
Farmer Jenny decides to expand her business interests and starts to package and sell the eggs produced by her chooks to a local shop.
The cost of producing $x$ dozen eggs per day is given by, in ...
2
votes
4answers
83 views
Solving a quadratic equation with precision when using floating point variables
I know how to solve a basic quadratic equation with the formula
$t_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
but I learned that if $b\approx\sqrt{b^2-4ac}$ floating point precision may give slightly ...
2
votes
2answers
69 views
quadratic equation
If $\alpha$ is root of equation $x^2+x+1 = 0$ then find the value of $1+\alpha +\alpha^2+\alpha^3+\cdots+\alpha^{2010}$
Here I have put the value of $\alpha$ in the given equation to get $1+\alpha + ...
2
votes
2answers
128 views
Quadratic Equation relation between roots
If the ratio of the roots of the equation $x^2+px+q=0$ are equal to the ratio of the roots of the equation $x^2+bx+c=0$ , then prove that $p^2c=b^2q$
Let $\alpha \& \beta$ be the roots of first ...
2
votes
1answer
86 views
Quadratic Equation using surds property
$$\left(\sqrt{2+\sqrt{3}}\right)^x+\left(\sqrt{2-\sqrt{3}}\right)^x=2^x$$
Using property of surd can we simplify the above expression like:
$$\left(\frac{\sqrt{3}+1}{\sqrt{2}}\right)^x ...
3
votes
1answer
52 views
Least value of $a$ for which at least one solution exists?
What is the least value of $a$ for which
$$\frac{4}{\sin(x)}+\frac{1}{1-\sin(x)}=a$$
has atleast one solution in the interval $(0,\frac{\pi}{2})$?
I first calculate $f'(x)$ and put it equal to $0$ to ...
4
votes
3answers
98 views
Values of $a$ for which $(a+4)x^2-2ax+2a-6 <0$ for all $x \in R$
How can we find all values of $a$ for which the inequality $(a+4)x^2-2ax+2a-6 <0$ is satisfied for all $x \in R$?
For the given condition, $D >0$, therefore $ (-2a)^2-4(2a-6)(a+4) >0$. ...
1
vote
1answer
66 views
quadratic equation - nature of roots
For what values of a does the equation $$x^2-( 2^a-1)x-3(4^{a-1}2^{a-2})=0$$ possess real roots?
Since the roots are to be real that means the discriminant should be $\geq 0$
$$\Rightarrow ...
0
votes
1answer
32 views
When finding the dilation factor of $y = 3(2x - 3)^2 - \frac{1}{4}$, why must the brackets be expanded?
When finding the dilation factor of $y = 3(2x - 3)^2 - \frac{1}{4}$, why must the brackets be expanded?
Why can't the outside factor of $3$ simply be used for the dilation factor from the ...
2
votes
3answers
107 views
Equation in the real world
Does a quadratic equation like $x^2 - ax + y = 0$ describe anything in the real world? (I want to know, if there is something in the same way that $x^2$ is describing a square.)
3
votes
0answers
70 views
Question about linearization
Given a data matrix $D\in\mathbb{R}^{N \times N}$
Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$,
if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
5
votes
2answers
95 views
Find the value of $x_1^6 +x_2^6$ of this quadratic equation without solving it
I got this question for homework and I've never seen anything similar to it.
Solve for $x_1^6+x_2^6$ for the following quadratic equation where $x_1$ and $x_2$ are the two real roots and $x_1 > ...
0
votes
0answers
47 views
Finding descent direction of quadratic function
I have a quadratic function: $f(x) = 24x_1+14x_2+x_1x_2$
and point $x_0 = (2,10)^T$ with $f(x_0) = 208$
And the first question is "give descent direction r in $x_0$"
The second question "is f convex ...
2
votes
2answers
85 views
Solving for the length of a side of a triangle
I have a problem in which I'm supposed to solve for the length of the two sides of the triangle below. I assumed that it would simply boil down to $x+5=\sqrt{4x+52}$, and converted to standard form, ...
