Questions about 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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### The maximum value of the smaller root of given quadratic function

Consider the quadratic expression : $$f(x) = x^2 +(a+2)x + (a^2 - a +2 )$$ given is that $a , p, q , (p<q)$ are real numbers and p and q are the roots of the equation $f(x)=0$. Q1) find the ...
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### Is there a way to solve a fourth grade equation if I know with which equations I got it?

So I have the multiplication of two quadratic equations that give me equal to a linear one, all with the same common variable. Am just trying to solve for that variable, but doing so with the ...
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### Quadratic convergence in numerical method

Let $$x_{n+1}= \dfrac{1}{2}\left ( x_n+\dfrac{R}{x_n} \right )$$ Interpret this relation in terms of quadratic convergence. State your condition for such quadratic convergence clearly. Prove that ...
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### Proving the existence of polynomial $p(x)$ and $q(x)$ such that $f(x)=1/2[p(x)+q(x)]$

Let $f(x)=x^2+ax+c$ where $a,c$ are real numbers. Prove that there exist quadratic polynomials $p(x)$ and $q(x)$ (with real coefficients) having all roots real such that $f(x)=\dfrac{1}{2}[p(x)+q(x)].$...
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### Simultaneous equations and tangrnts with unknown value $k$ and tangent [closed]

The line with equation $2x+3y= k$ s a tangent to the circle $x^2+y^2+6x+4y=0.$ Find two possible values of $k$
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### Solve $\begin{cases}x^2=7x+3y\\y^2=7y+3x\end{cases}$ [closed]

Solve $$\begin{cases}x^2=7x+3y\\y^2=7y+3x\end{cases}$$ Can you give me a hint? $(0;0)$ and $(10;10)$ are obviously solutions. I can't see how to approach the problem, though.
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### $f(x)$ is a quadratic function with vertex $(1, −2)$, opens up. [closed]

I got this on a practice quiz. I couldn't figure it out without at least another point on the graph. $f(x)$ is a quadratic function with vertex $(1, −2)$, opens up.
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### A quadratic $ax^2+bx+c$ has its roots in the interval $[0,1]$, find the maximum value of $\frac{(a-b)(2a-b)}{a(a-b+c)}$

A friend of mine sent me this problem asking for help, however, I myself, need help, so here am I. I was able to find the minimum of the denominator, which was equal to $b^2/a$ if my steps were ...
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### how to find $\frac{a}{b}$ given that $2a^2 + 2007a + 3 = 0$ and $3b^2 + 2007b + 2 = 0$

Given that $$\begin{cases} 2a^2 + 2007a + 3 = 0 \\ 3b^2 + 2007b + 2 = 0 \end{cases}$$ and $ab \ne 1$, how to solve for $\frac{a}{b}$? My try: \begin{align} 2007a = -3 - 2a^2 \\ 2007b = -2 - 3b^2 \\ \...
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### How to solve this question of algebra logically? [closed]

solve this. If $x ^ 2 - x ( 6 - \sqrt 3 ) + 10 - 3 \sqrt 3 = 0$, then find $$( x + 3 ) ^ { 17 } + \frac 1 { ( x + 3 ) ^ { 17 } } \text .$$
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### Finding $\sum_{n=1}^{\infty} \frac{1}{f(n)}$ where $f$ is a real quadratic function?

Let's consider the following function $f(x)=ax^2+bx+c$, where $a, b$ and $c$ are all real constants. Is there any way to calculate the value of $$\sum_{n=1}^{\infty} \frac{1}{f(n)}\text{?}$$ I can ...
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### For any real number c, the quadratic equation $x^2+x-c^2 = 0$ has two distinct (real) solutions. Is this true or false and explain why.

I am a first year math major taking the introductory proofs course. This is my solution to the question. I would like you to check if my solution is correct or complete. The statement is true. In ...
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### Quadratics - Nautical miles and knots question

Ship A is 50 nautical miles west of Ship B. Ship A is heading east at 10 knots and ship B is heading south at 5 knots. Find the minimum distance between the ships, and at what time it occurred What ...
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### Solve the equation $z^3-2z^2+3z-2=0.$ [closed]

Solve the following equation. $$z^3-2z^2+3z-2=0$$ If $a$ is a complex solution of this equation, what does $A$ equal? $$A= \frac{|a|^2}{1-i ^ {43}}$$ It's on my exams and I really need to solve this ...
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### Solving a trigonometric equation for purely imaginary numbers [closed]

I'm puzzled with the following exercise: "By constraining $z$ to be purely imaginary, show that the equation $\cos{(z)}= 2$ can be represented as a standard quadratic equation. Solve this ...
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### How to prove the following inequality $x+y\ge2$

Let $x$ and $y$ be two real positive integers, such that: $x+y+xy=3$ prove that $x+y\ge2$ I tried some simplifications like this one $x(1+y)=3-y$ and $y(1+x)=3-x$ and using the fact that both of $x$ ...
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### Solving a quadratic equation with 3 parameters [closed]

me and my group of students are having trouble solving the following quadratic equation. Any help is appreciated. Thanks in advance.
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### What is the Product of the roots of the first equation?

Both of the following equations have real roots. $$ax^2 +bx+c=0$$ $$(a-b+c)x^2 -2(a-c)x+ (a+b+c)=0$$ If roots of the second equation are α and β show that $\frac{(1-α)(1-β)}{(1+α)(1+β)}$ is the ...
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### How to find the parameter b such that the following sum of quadratic expressions is minimized?

Suppose you have $x_1, ..., x_n$. My task is to find $b \in \Bbb{R}$ such that the sum $\sum_{i=1}^n (x_i - b)^2$ is minimal. Now, I think we can view it as a multivariate function and differentiate.....
If $n$ is a constant and if there exists a unique value of $m$ for which the quadratic equation $x^2 + mx + (m+n) = 0$ has one real solution, then find $n$. Let the roots of the quadratic be $r,s.$ ...