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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0
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2answers
29 views

How to prove a solution of equation is rational if another one is rational number?

The question is : $r$ is the solution of equation $x^2+bx+c=0$ and $r$ is a rational number, so there is another solution $s$, how to prove s is a rational number as well? I have no idea about it and ...
0
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2answers
27 views

Maximums on Quadratic Functions [on hold]

How do you find the maximum of a quadratic function? Specifically, $R(x) = -4x^2 + 4000x$
0
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0answers
12 views

Disadvantages of Taylor series method

There is method called Taylor series method to solve non linear equations iteratively. I am interested to know ,what are the disadvantages of using this method to solve. General Idea any one please?
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2answers
27 views

I don't understand how the following algebraic equation breaks down. I just can't figure out how this answer is devised.

I just don't understand how this equation breaks down like this. The second step... $[k^2 + k + 2k + 1]$ is perplexing, but breaking that down to $(k+1)(k+2)$ has completely baffled me. I would expect ...
3
votes
5answers
63 views

Why is the solution to $\sqrt{6-5x}=x$ only $x=1$ and not $x=-6$? [duplicate]

I solved the equation $\sqrt{6-5x}=x$ as follows: $$(\sqrt{6-5x})^2=x^2$$ $$6-5x=x^2$$ $$0=x^2+5x-6=(x+6)(x-1)$$ $$x=-6 \quad \text{or} \quad x=1$$ If I plug in $x=-6$ into the original equation, I ...
1
vote
1answer
36 views

How to find $f(2)+f^{-1}(5)$ if $f(2x^2+3x+4)=6x^2+9x+20$? [on hold]

$$f(2x^2+3x+4)=6x^2+9x+20$$ How to solve $f(2)+f^{-1}(5)$ ? Any help or advice on solving is much appreciated. Thanks!
4
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2answers
40 views

Sign of determinant of a $3 \times 3$ matrix with entries $1+\alpha^{i+j-2}+\beta^{i+j-2}$, for distinct $\alpha,\beta\in\mathbb R\setminus\{1\}$

Let $ \alpha\ne1,\beta\ne1$ be the distinct real roots of the equation $$ax^2+bx+c=0,~~a,b,c\in \mathbb{R},a\ne 0$$ Let $S_n=\alpha^n+\beta^n,n\geq0$ and ...
0
votes
1answer
26 views

The nature of roots of the quadratic equation $ax^2+(b-c)x-2b-c-a=0,$

The expression $ax^2+2bx+c$ where $a$ is a non-zero real number,has the same sign as that of $a$,for every real value of $x$,then roots of the quadratic equation $ax^2+(b-c)x-2b-c-a=0,$are $(A)$real ...
0
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1answer
34 views

two positive real numbers have their sum, product… [closed]

Two positive real numbers have their sum,product and difference of the squares $(a^2-b^2)$ equal. Find those numbers. It would be easy to solve if only two of these were mentioned, but I don't know ...
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2answers
57 views

is there a triangle with sides $2,3,6$?

Is there a triangle with $a=2, b=3, c=6$? (I know there's not because sum of any two sides has to be greater than the third side) How much do we need to extend these sides to get a right triangle ...
1
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3answers
29 views

solving rectangle

Diagonal of a rectangle is $13$ cm. If we extend the length of the rectangle for $4$ cm and width for $7$ cm, then diagonal will be longer for $7$ cm as well. Find sides (length and width) of the ...
0
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2answers
27 views

speed and travel problems

The ship left the harbor and is travelling $60$ kilometers downstream. Then it continues on the tributary river (upstream) $20$ kilometers. Travel took $7$ hours to finish. River speed is $1$ km/h. ...
1
vote
1answer
19 views

Roots of polynomials combined with Trigonometric Functions

If $$ f(x) = x^2 + ax + d \cos x $$, where $a$ is an integer and $d$ is a real number, what are all possible values of the tuple $(a,d)$ such that $f(x)$ and $f(f(x))$ have the same set of real roots? ...
1
vote
2answers
28 views

How to derive in a quadratic equation. [closed]

I was reading Purplemath's lesson about quadratic equations, and came to the part about deriving the solution to $ x^2 + 6x + 10 = 0$. I understood the part about putting the loose number in the other ...
1
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3answers
59 views

Find $\alpha^3 + \beta^3$ which are roots of a quadratic equation.

I have a question. Given a quadratic polynomial, $ax^2 +bx+c$, and having roots $\alpha$ and $\beta$. Find $\alpha^3+\beta^3$. Also find $\frac1\alpha^3+\frac1\beta^3$ I don't know how to proceed. ...
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3answers
50 views

What comes first here? pemdas doesnt really tell me what to do here

So I have this equation: $2x(x+3)(x+3)$ Do I FOIL the $(x+3)$ first or multiply the $2x$ to the first $(x+3)$? Would there be a difference? Isn't multiplication commutative?
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1answer
9 views

Writing the function to maximize volume or a cylinder

A rectangular piece of paper is curled into a cylinder with two open circles on each side. The perimeter of the piece of paper is 124 inches. What is a function that could be written to find the ...
2
votes
4answers
72 views

If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$

If $x^2+3x+5=0$ and $ax^2+bx+c=0$ have a common root and $a,b,c\in \mathbb{N}$, find the minimum value of $a+b+c$ Using the condition for common root, $$(3c-5b)(b-3a)=(c-5a)^2$$ ...
1
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1answer
26 views

Triple Simultaneous Equations not resolving the Equation for a Quadratic Function

so I'm doing this math problem for my Calculus I course in college. Here is a screenshot of the problem: Graph 1 (click here to view); the prompt is "Find an expression for the quadratic function ...
4
votes
3answers
57 views

Is this quadratic pointing up or down? How do I know?

The equation is $-2x^2 + 4x + 30 = 0$. I simplified it to $-2(x^2 - 2x - 15)$. To know if it points up, I need to look at $ax^2$, and if $a > 0$ it is up and if $a$ is $< 0$ it is down. ...
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3answers
28 views

Which form of this quadratic do i use to solve intercept and range?

So my equation is: $-2x^2 + 4x + 30 = 0$ If I use this form to look at my y intercept, it will be 30. However, once I simplify it to: $x^2 - 2x - 15$, then my y intercept will be $-15$. Which one do ...
1
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3answers
25 views

Which form do I use in this quadratic example for factoring? [closed]

Alright, so I have the equation: $x^2 + 5x = -4$ To find the axis of symmetry and the zeroes (by using the quadratic formula), would I have to write it in standard form, which is $x^2 + 5x + 4 = 0$, ...
0
votes
1answer
36 views

Determine the equation of a parabola with roots $2 + \sqrt {3}$ and $2 - \sqrt {3}$, and passing through the point $(2,5)$

My attempt: $$f(x) = a(x - r)(x - s)$$ $$f(x) = a(x-(2 + \sqrt {3}))(x-(2- \sqrt {3}))$$ From here, I'm stuck. I can't remember where to go from this point and need some help.
2
votes
1answer
45 views

Writing an equation from data

Wendy, Elizabeth, and Charlie are all working on a math problem together and they are having a disagreement.: Ticket lines are huge at the Math Olympics ticket office. Pi, the local math team, is ...
0
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1answer
21 views

Why Non Linear equations put equal to zero in Newton Raphson Mehotd

While solving non linear equations we put them equal to zero in Newton-Raphson Method.Why we do that? Any Idea?
2
votes
2answers
83 views

Are $\mathbb{C}^2$ and $\mathbb{C}^2/(x,y)\sim(y,x)$ homeomorphic?

Let $A$ be the set of monic quadratics over $\mathbb C$ and let $B$ be the set of unordered pairs over $\mathbb C$ where possibly the two elements of the pair may be the same. Then the map which takes ...
2
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2answers
31 views

Solving a radical equation with trinomials on both sides

$$8\sqrt{a^2-4a-16}=3a^2-12a-64$$ I do know the standard procedure—square both sides, isolate square root, square again, check solutions to make sure they are real, etc. However, for a problem such ...
0
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1answer
30 views

Solve equation for t

$$s = 2 \ln|\tan(t) + \sec(t)|$$ I tried to solve it and got a quadratic equation which turned out to equal $arcsin(\dfrac{-2 \pm e^s}{2(1+e^s)})$ This doesn't seem right. Any thoughts?
0
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3answers
64 views

Find all rational points where $x^2 - y^2 = 1$ (need help simplifying quadratic formula) [duplicate]

The original problem is to find all rational points where $x^2 - y^2 = 1$ I know how to go about the problem, but whenever I get to the point of simplifying my equation, I keep having problems. This ...
1
vote
1answer
25 views

Quadratic function that produces natural number from natural number inputs

I am currently trying to find a way to generate different (preferably quadratic) function as part of a encryption algorithm such that : ...
1
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2answers
41 views

Roots of $f(x)+g(x)$

Question : Let $p,q,r,s \in \mathbb R$ such that $pr=2(q+s)$. Show that either $f(x)=x^2+px+q=0$ or $g(x)=x^2+rx+s=0$ has real roots . My method : To the contrary suppose that both $f(x)$ and ...
0
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3answers
52 views

Why not always use the quadratic equation

The is a very simple question, but I have just started studying quadratics. I understand how to factor them using different methods and also understand solving a quadratic using the formula, but my ...
1
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1answer
40 views

Find integers $x$ and $y$ such that $\frac{27^{x+y}}{9^{xy}}=27$ and $\frac{4^{2xy}}{8^{x+y}}=512$ .

Find all the integers $x$ and $y$ such that : $$\frac{27^{x+y}}{9^{xy}}=27$$ and :$$\frac{4^{2xy}}{8^{x+y}}=512$$ I'm in Algebra two and I feel like there are certain types of math I haven't ...
2
votes
3answers
75 views

How can you find $m$ in $mx^2+(m-3)x+1=0 $ so that there is only one solution

How can you find $m$ in $$mx^2+(m-3)x+1=0 $$ so that there is only one solution. I tried to solve it by quadratic equation but I end up with two solutions. So I want it know that is there a way so ...
0
votes
1answer
91 views

Can this nonlinear simultaneous equations be solved?

Problem: Can this nonlinear simultaneous equations be solved about $\mathbf{x}$? Then $\{A,B,E\}\in R^{n \times n}$ are symmetric matrices, $\{\mathbf{x},\mathbf{y}\}\in R^{n}$. Particularly, $E$ is ...
0
votes
2answers
64 views

Solve the following number theory problem with 2 variables [closed]

Let there be $$a,b∈ \Bbb Z$$ Demonstrate that there exist no solutions for the following equation $$a^2-3b^2=-1$$
1
vote
1answer
84 views

How do the roots of “$x^2 + bx + c$” change as $b$ is kept constant and $c$ is changed? [closed]

Consider the function $x^2 + bx + c$ How do the (real or complex) roots of the equation change if $b$ is held constant and $c$ is changed? I.e. Which patterns are evident? What would it look like if ...
0
votes
0answers
14 views

Show that $p(x)=rq(x)$ for some rational number $r$.

Let $p(x)$ and $q(x)$ be two quadratic polynomials with integer coefficients. Suppose they have a non rational root in common. Show that $p(x)=rq(x)$ for some rational number $r$.
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2answers
31 views

For odd integers $a,b,c$ line $ax+by+c=0$ cannot intersect parabola $y=x^2$ in a rational point

For odd integers $a,b,c$ line $ax+by+c=0$ cannot intersect parabola $y=x^2$ in a rational point(where both abscissa and ordinate are rational numbers.) We need to solve the equation of the line ...
37
votes
12answers
5k views

Why do I get one extra wrong solution?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by (-1): $$\sqrt{x}=x-2$$ power of 2: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $1$ is not a ...
2
votes
2answers
54 views

How do I factorise this difficult quadratic without a calculator?

On one of the UKMT maths challenge past papers(Team challenge) it asks you this question: Factorise $120x^2 + 97x - 84$ That is the whole question. I used a calculator and found that you factorise ...
1
vote
3answers
67 views

Solve this complex number: $z^2+(1+i)z+i=0$

I need to solve: $$z^2+(1+i)z+i=0$$ first of all I used $$z = (a+ib)$$ and I get: $$(a+ib)^2+(1+i)(a+ib)+i=0$$ $$a^2-b^2+2aib+a+ib+ia-b+i=0$$ then I have ordered, on the left the number without ...
1
vote
2answers
44 views

Why are these 2 algebraic expressions equivalent?

I just solved a long problem for my physics w/calculus homework that required a simplification using a quadratic formula. The "textbook" (flipItPhysics) came up with a different simplification than ...
0
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4answers
39 views

Determine if there is a solution in inequality [closed]

Does this inequality have a solution? $$x^2 < 5x - 6$$ I try to solve but after I check, the solution is wrong.
3
votes
1answer
31 views

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$ As $r$ is the integer root of the equation ...
0
votes
1answer
28 views

Calculate the quadratic residues in Z∗17.

Hello I am wondering if any one can help me I am trying to figure out how these below answers where came to too. Calculate the quadratic residues in Z∗17. Solution: This can be done by direct ...
0
votes
0answers
115 views

Question about an equation using linear algebra

Firstly, in $n \in \bf{N}$, we consider the square matrices as $\bf{\{H_{i},A_{i},P,Q \}} \in \bf{R}^{\it{n \times n}}$ and the vectors $\bf{\{ a,x,1 \}} \in \bf{R}^{\it{n}}$. All parameters are known ...
6
votes
2answers
119 views

Looking for a function $g(x)$ such that $g(2x+2) = g(x) + 2x+2$

So recently I got bored in maths class (I'm in tenth grade) and made up a little equation that looked something like this: $$g(f(x)) = g(x) + f(x) $$ My original goal was to find different $g(x)$ to ...
0
votes
3answers
84 views

No. of parabolas possible for the given equation.

Given, $f(x)=-x^2+qx+r$. $(q,r) \epsilon R$. $q,r$ are variables. A quadratic equation $f(x)=0$ has a maximum value $m$ ($m$ is a constant) and a root $x=a$. Does $f(x)$ correspond to a unique ...
3
votes
1answer
38 views

Easy word problem but am I working it too hard?

A friend of mine gave me this problem: A man who walks at a constant speed goes to his barn 30 miles away with a 2 mph wind pushing against him. After arriving at the barn he remembers he ...