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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For example,we need to find the range of a for which expression $\dfrac{ax^2+3x-4}{3x-4x^2+a}$ assumes all real values for real values of $x$ how to proceed?
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### What is the equation for this circle? [closed]

Circle C is plotted on a graph. Circle C's center is (0,0). A tangent of circle C goes through the points (0,10) and (-30,0). What is the equation for the circle and how do you calculate this? Circle ...
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### Solving a quadratic equation problem with two variables

This is a post of two three problems regarding the method to solve bivariate quadratic equations. In brief, How does the elimination happen here. Or, how is the elimination used? (Update, I know how ...
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### Using the discriminant to find the value of k.

Question:- $x^2-4x-1=2k(x-5)$ has two equal roots. Calculate the possible values of $k$. I know that that must mean the discriminant must equal $0$. So I found: $b = (-2k-4)$ $a = 1$ $c = (10k-1).$ ...
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### Finding the inverse function of $f(x) = -(9x^2 + 6x + 2)e^{-3x}$ using the generalised Lambert function.

I just learned about the generalized lambert function and I was trying to use it to find the inverse of a function. I have solved the equation below, so can someone check if it is correct? Thanks in ...
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### How do you approach when completing the square?

If $M = 3x^2 - 8xy + 9y^2 - 4x + 6y + 13$, where $x,y\in\mathbb R$, then $M$ must be: a) positive $\qquad$b) negative $\qquad$c) $0 \qquad$ d) an integer I somehow managed to figure it out by ...
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### Graph of quadratic $f(x)=ax^2+bx+c$ when $a$ is fixed and $b,c$ are varied

I noticed a small thing while playing with the graph of quadratic. $$ax^2+bx+c = a\left(x+\frac{b}{2a}\right)^2 + c - a\left(\frac{b}{2a}\right)^2$$ Clearly $b,c$ only determine how the vertex of the ...
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### Would an equation in the form ax + b = c/x be considered quadratic?

For example is $2x + 3 = \frac 5x$ quadratic? On the one hand it has two solutions, $x = 1$ and $x = -5/2$ which is the number of solutions we'd expect from the fundamental theorem of algebra but on ...
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### Sum of squares of consecutive integers equals to a square [duplicate]

I am not at all mathematics guy, just had a question. How can I find possible pairs of consecutive integers whose sum of squares equals to a square? I understand equation will be something like: x² + (...
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### Mistake in MIT paper? [closed]

This is MIT paper: https://ocw.mit.edu/courses/physics/8-01sc-classical-mechanics-fall-2016/readings/MIT8_01F16_chapter14.9.pdf Look at equations $(14.8.26)$ and $(14.8.27)$. Did I forget how to solve ...
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### design the equation of a decreasing parabola (between 2 points)

I am trying to come up with equations that can create a smooth downward decrease along $y$ so it falls from $0.997$ down to $0.990$. I am interested at the interval [N/5, N] along the $x$-axis. I ...
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### Factoring the quadratic equation $3x^2-23x+14=0$

I'm having trouble understanding how to factor this equation. Let's go step by step: First I use the sum/product pattern: $$3x^2−2x−21x+14=0$$ Then I take the common factors: $$x(3x−2)−7(3x−2)=0$$ ...
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### Why is this approach wrong?

If $\alpha$ and $\beta$ are the roots of $x^{2} - 4 x - 3$, then find $$\frac{1}{(\alpha-4)^{2}}+\frac{1}{(\beta-4)^{2}}$$ Solution This is the approach that gives wrong answer. \begin{array}{l} x^{2}...
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### Why does Solving system of quadratic equations gives extra roots?

Consider these system of Equations \begin{align*} \begin{cases} x^2+4x+4=0\\\\ x^2+5x+6=0 \end{cases} \end{align*} For solving them We have Method 1- Subtract both equations So $-x-2=0$ Hence, $x=-2$ ...
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