# 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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### Solution of given equation for $x$

Solve the given equation for $x$ $$\sqrt{x^2-2x+8}+\sqrt{x^2-2x+3}=125$$ I solved the question by taking ${x^2-2x+3}=t$, and squaring twice and finally solving ${x^2-2x+3}=t$ but it required very ...
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### Solve the following $\frac{3x}{x+6} \ge 0$

Solve $$\frac{3x}{x+6} \geq 0$$ My work $$(x+6) / 3x <0$$ $$1/3 + 6/x <0$$ $$6/x <-1/3$$ $$x >-18$$ is that correct
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### Solving $x^2 - 16 x+55> 0$ for $x$

Solving $x^2 - 16 x+55> 0$ for $x$ my work $$(x-11)(x-5) > 0$$ then x >11 and x > 5 is that correct ???
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### For $ax^2+bx+c$ prove that $|a|+|b|+|c|\leq 17$

Let $ax^2+bx+c$ be a quadratic polynomial with real coefficients such that $$|ax^2+bx+c| \leq 1,$$ for $0\leq x\leq 1$. Prove that $$|a|+|b|+|c|\leq 17$$ How to proceed in this particular question. ...
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### Finding integer solutions to quadratics in the form [duplicate]

In a set containing two different types of elements the probability of randomly choosing two elements of the same type can be expressed as: $$\ \frac nm * \frac {n-1}{m-1} = \frac 1x$$ Where n is ...
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### Are there more quadratics with real roots or more with complex roots? Or the same?

Consider all quadratic equations with real coefficients: $$y=ax^2+bx+c \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,, a,b,c \in \mathbb{R}, \, a≠0$$ I was wondering whether if more of them have real roots, more ...
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### Vertex (smallest possible value) of $ax^2+bx+c$

The original problem was this: Find the smallest possible value of $ax^2+bx+c$, where a, b, and c are given numbers and $a>0$, and x is some number. I already asked this, and got a decent answer, ...
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### quadratic function vs conic section

I am categorizing types of math problems on the ACT. I started off with 'quadratic function' as one category, and 'conic sections' as another... It seemed like a simple classification at first, but ...
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### Re-write a quadratic equation in another form?

$x^2 + \sqrt{2}x = \frac{1}{2}$ I need to find the real solutions for this equation and write it in this form: $$\frac{-\sqrt{A} \pm B}{C}$$ So when I work the problem out with the quadratic ...
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### How can I imagine double/repeated root of a quadratic equation?

A quadratic equation such as $(x-2)^2=0$ has a repeated root $(2,2)$. A lot of things in math (equations, matrixes and so) can be nicely drawn on a $2D, 3D$ etc plane (with $x$-$y$ axis). I mean, I ...
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### Numerical scheme approximate integral.

enter image description here Hi guys the question is inside the image. for Q(a): (i): My idea is w1=w2=(a+b)/2 because of trapezoid rule. Am I correct? (ii):Need help. (iii): Should I follow the ...
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### Quadratic equation solutions modulo prime p

the question is: find all primes p that satisfy the equation: x^2-2*x-5 = 0 (mod p) The discriminante is 24, and I know that the equation mod p has a solution ...
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### Smallest possible value of $ax^2+bx+c$

The problem goes like this: Let $a, b$ and $c$ be given numbers, where $a>0$, and let $x$ be some number. What is the smallest possible value of $ax^2+bx+c$ ? The terms 'given number' and 'some ...
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### Solution of equation

If $f(x) = x^2 - 2ax + a(a+1)$ , $f:[ a, \infty] \to [a,\infty]$ . If one of the solution of the equation $f(x)=f^{-1}(x)$ is $5049$ , then what may be the other solution ? My WORK: I found the ...
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### location of roots of quadratic with natural coefficients

the quadratic equation $ax^2-bx+c=0$ ; $a,b,c \in \mathbb{N}$, has two distinct real roots belonging to the interval $(1,2)$ , then what would be least value of $a$ and $b$? I tried to solve these ...
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### For what values of $a,b,c$ will $ax^2+bx+c \geq 0$ hold $\forall x \in \mathbb{R}$?

If I let $y=ax^2+bx+c, (a\neq 0)$ then extremum of $y$ is attained at $x=-\frac{b}{2a}$. Then $\large\frac{\mathrm {d^2}y}{\mathrm {d}x^2}\big|_{(x=-\frac{b}{2a})}=2a$ which is positive or negative ...
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### How to solve for log with a number outside?

$$\log_6(4x-10)+1 = \log_6(15x+15)$$ This is a sample problem. I know that when the bases of log are the same, all you have to do is set the parenthesis inside equal to each other. If the $1$ wasn't ...