# Tagged Questions

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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### How to solve a quadratic inequality that acts like a quadratic equality?

This will be largely a trivial question. But how do I solve an inequality like this: $3x^4 - 4x^2 + 1>0$ ? Of course, I can treat it like a quadratic inequality by saying $t=x^2$ So I can solve ...
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### If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then the value of m is equal to to?

If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then the value of m is equal to to ? I drew the graph of $|x^2+4x+3|$.I found that that for the given condition $mx-2m$ must be a ...
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### If $\alpha,\beta$ are roots of $x^2+px+q=0$ and also of $x^{2n}+p^nx^n+q^n=0$

If $\alpha,\beta$ are roots of $x^2+px+q=0$ and also of $x^{2n}+p^nx^n+q^n=0$ and $\frac{\alpha}{\beta}$,$\frac{\beta}{\alpha}$ are the roots of $x^n+1+(x+1)^n=0$, then $n$ is Odd Even Any ...
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### 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 ...
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### Maximums on Quadratic Functions [closed]

How do you find the maximum of a quadratic function? Specifically, $R(x) = -4x^2 + 4000x$
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### 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 ...
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### 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 ...
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### How to find $f(2)+f^{-1}(5)$ if $f(2x^2+3x+4)=6x^2+9x+20$? [closed]

$$f(2x^2+3x+4)=6x^2+9x+20$$ How to solve $f(2)+f^{-1}(5)$ ? Any help or advice on solving is much appreciated. Thanks!
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### 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 ...
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### 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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### 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 ...
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### 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.
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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 : ...
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### 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 $g(x)$...
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### 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 ...
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### 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 ...
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### 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 ...
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### Can this nonlinear simultaneous equation be solved?

Problem: $\{A,B,E,R,S\}\in R^{n \times n}$ are square matrices, $\{\mathbf{x},\mathbf{y}\}\in R^{n}$ are vectors. Particularly, $\{ A,B \}$ are symmetric matrices, and $E$ is an identical matrix. We ...
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### 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 $$a^2-3b^2=-1$$
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...