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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1answer
23 views

Find all values of parameter a, when sum of solutions of following equation is 100

Find all values of parameter $a$, when sum of solutions of following equation is $100$. $$ \sin(\sqrt{ax-x^2})=0 $$ I tried to get rid of that $sin$ and there was quadratic equation with two ...
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1answer
14 views

Help with some calculations

My question is: what I need to do to get 2nd equation from the first? 1) $TP1 = vp1 · λ + TS1$ $TP2 = vp2 · λ + TS2$ 2)$$TP_2 − TS_2 =\frac{vp2}{vp1}(TP1 − TS1)$$
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3answers
61 views

Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k$

A quadratics question. Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k.$ I recently asked a similar question, but this problem seems ...
1
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1answer
52 views

For what values does $kx^2 - 6x - 4 = 0$ cut the x-axis? [on hold]

A quadratic equation problem. For what values does $kx^2 - 6x - 4 = 0$ cut the x-axis? The preceding part of this question dealt with the discriminant.
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1answer
29 views

If the volume of a container is 196 cm^3, find the dimensions of the original template.

This is a quadratics problem. The full question reads: An open container with a square base is made by cutting 4 cm square pieces out of a piece of tin. If the volume of the container is 196 cm^3, ...
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5answers
78 views

The roots of the equation $x^2 - 6x + 7 = 0$ are $α$ and $β$. Find the equation with roots $α + 1/β$ and $β + 1/α$.

Quadratic equation question, as specified in the title. The roots of the equation $x^2 - 6x + 7 = 0$ are $α$ and $β$. Find the equation with roots $α + \frac{1}{β}$ and $β + \frac{1}{α}$. I ...
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1answer
39 views

Limit of the root of quadratic equation

The root of the equation $ a x^2 + bx + c = 0 $ is given by $$ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \;\;\;...(1) $$ On the other hand, if $a = 0$, then from the original equation we get $$ x = - \...
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6answers
60 views

How to solve without solving by inspection? $\frac{x+5}{x+k}=\frac{-kx+5}{x-1}$

Background: This is from a test review on functions. The original problem was Find the value of $k$ so that the function $f(x) = \frac{x+5}{x+k}$ will be its own inverse. I found the answer by ...
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2answers
56 views

Solve the quadratic equation

$$ \sqrt{a-\sqrt{a+x}}=x $$ This equation contains one variable x we have to find the value of x.I tried to simplify it but it doesn't work....i have also tried the basic concepts of quadratics ...
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2answers
44 views

Quadratic inequality proof.

I recently encountered a quadratic equations property that $ax^2+bx+c>0$ $ \forall$ $x\in \Re \Rightarrow D<0$ $and$ $a>0$ and $ax^2+bx+c<0$ $ \forall$ $x\in \Re \Rightarrow D&...
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1answer
34 views

System of two Nonlinear equations

I have a probably very simple problem here. A system of nonlinear equations. $$\left\{ \begin{align} & {{x}^{2}}+{{y}^{2}}=26 \\ & x+{{y}^{2}}=6 \\ \end{align} \right.$$ I started with ...
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1answer
65 views

Permuting the roots of a cubic polynomial with a quadratic polynomial cyclicaly

The polynomial $Q(x)=x^3-21x+35$ has three distinct real roots $r,s,t$. Find reals $a,b$ so that $P(x)=x^2+ax+b$ satisfies $P(r)=s,P(s)=t,P(t)=r$ or $P(r)=t,P(t)=s,P(s)=r$. I tried using cardano to ...
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2answers
46 views

how to solve a trignometric quadratic equation?

I am stuck in a question... the questions says $sin^4 x -(k+2)sin^2 x -(k+3)=0$ has a solution then what is the interval in which k must lie. I tried to solve it by putting $sin^2 x =p$ then putting ...
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1answer
54 views

What is the equation of the quadratic function through $(2,5)$ with roots $1+\sqrt 5$ and $1-\sqrt 5$? [on hold]

Determine the equation of the quadratic function that passes through $(2,5)$ if the roots of the corresponding quadratic equation are $1+\sqrt 5$ and $1-\sqrt 5$.
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2answers
81 views

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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5answers
77 views

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
2
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3answers
57 views

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 ???
5
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2answers
79 views

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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0answers
28 views

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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1answer
55 views

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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1answer
20 views

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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1answer
36 views

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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1answer
15 views

The solution set of a multivariate quadratic after transformation of variables

I have an equation given in terms of three vectors $\vec{x}$, $\vec{y}$ and $\vec{z}$, all in $\mathbb{R}^n$: $$1 - (\vec{x}\cdot\vec{y})^2 = K + 2K\vec{z}\cdot\vec{y} + (\vec{z}\cdot\vec{y})^2, K \...
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1answer
35 views

How can I find the largest possible subset A of $\mathbb{R}$?

So I have this equation $$f(x)= \frac{x^2}{(x-2)(x+3)}$$ and I need to find the largest possible subset $A$ of $\mathbb{R}$ that could form the domain of a function. Can anybody help me? I really don'...
9
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1answer
54 views

If $a$ and $b$ be the roots of the quadratic equation $x^2-6x+4=0$ then find the value of given expression.

Let $a$ and $b$ be the roots of the quadratic equation $x^2-6x+4=0$ and $P_n = a^n + b^n$ then the value of $$\frac{P_{50}(P_{48}+P_{49})-6P_{49}^2+4P_{48}^2}{P_{48}.P_{49}}$$ Options are $(A)$ $2$ ...
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4answers
55 views

When to simplify a quadratic equation?

I had the following quadratic equation: $$38x^2 - 140x - 250 = 0$$ And before starting to solve it, I simplified it by dividing all terms by $2$: $$19x^2 -70x - 125 = 0$$ But when I solved it I got: $...
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2answers
39 views

Determine if quadratic diophantine equation in two variables will generate perfect squares

I have come across two equations with variables $x,y$ \begin{align*} (x+ay)^2+ 4 x y\\ (x-y)^2-4 c x y \end{align*} where $a,c\in \mathbb{Z}_+$ are some constants. I would like to determine the ...
3
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2answers
64 views

How to get the coordinates of the center of the ellipse after approximation

I create an algorithm recognizing ellipses in images. I have five coordinates (points) possible ellipse. (8.8) (7.4) (6.3) (3.6) and (2.2) I use the formula of the conical section of the ...
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0answers
14 views

Solution for quadratic equation for Eikonal Equations.

H. Zhao states that the unique solution to the equation $$ [(x-a)^+]^2 + [(x-b)^+]^2 = f^2 h^2 $$ is $$ \bar{x} = \left\{ \begin{matrix} \min(a,b) + fh &|a-b| \ge fh \\ \frac{1}{2} \left(a + ...
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2answers
515 views

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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2answers
45 views

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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0answers
13 views

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 ...
2
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2answers
34 views

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 ...
0
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3answers
60 views

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 ...
3
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1answer
32 views

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 ...
3
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1answer
35 views

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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2answers
46 views

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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2answers
50 views

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 ...
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1answer
32 views

Largest equation of a circle that shares 2 tangents with a curve

Just played around on a graphic calculator a little, and discovered that given the curve $y=x^2$ , all circles with the equations in the form of $\left(y-a\right)^2+x^2=\frac{4a-1}{4}$ for all $a>0....
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1answer
17 views

How to find quadratic vertex form function given a point?

Write an equation of a parabola that has a directx of y= -5 and a focus at (2,-1)? I'm guessing focus here means the vertex $$ Y = a(x-h)^2 + k$$ $$-5 = a(x-2)^2 -1$$ $$-5 + 1 = a(x-2)^2$$ If i ...
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2answers
33 views

A system of polynomial equations of degree $2$ in two variables

I need to find an explicit solution of this system of polynomial equations of degree $2$ in two variables $x,\,y$: $$\begin{cases} p_1x^2+q_1y^2+r_1xy+s_1x+t_1y+u_1=0\\ p_2x^2+q_2y^2+r_2xy+s_2x+t_2y+...
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1answer
33 views

$5$ is quadratic residue mod $p$ if and only if $ p\equiv \pm 1, \pm 9 \pmod {20}$

5 is quadric residue mod p if and only if $ p\equiv +/- 1, +/-9 \pmod {20}$ $$(5/p)=(p/5)$$ $p\equiv 1 \pmod 4$ ⟹ $1,5,9,13,17 \pmod {20}$ $p\equiv 1 \pmod 5$ ⟹ $1,6,11,16 \pmod {20}$ then $p\...
0
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1answer
56 views

How to solve something like $x^2 + \sqrt{2}\,x - 3$?

What is exactly the MO when it comes to solving a quadratic equation like $x^2 + \sqrt{2}\,x - 3$? Do I take the part with the under root to the other side and end up with $x^4$?
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2answers
52 views

Equation of a tangent on a circle given the gradient and equation of the circle

My maths teacher told me this problem was impossible without knowledge of implicit differentiation: is she right? You are given the equation of the circle $\left(x+2\right)^2+\left(y-2\right)^2=16$ , ...
5
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2answers
74 views

Condition on $a$ for $(x^2+x)^2+a(x^2+x)+4=0$

Find the set of values of $a$ if $$(x^2+x)^2+a(x^2+x)+4=0$$ has $(i)$ All four real and distinct roots $(ii)$ Four roots in which only two roots are real and distinct. $(iii)$ All four imaginary ...
3
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1answer
67 views

Finding all pairs of integers that satisfy a bilinear Diophantine equation

The problem asks to "find all pairs of integers $(x,y)$ that satisfy the equation $xy - 2x + 7y = 49$. So far, I've got \begin{align} xy - 2x + 7y &= 49 \\ x\left(y - 2\right) + 7 &= 49 \\ ...
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3answers
51 views

Equation of a tangent that passes through a point not on the curve

Just made this little question for me and a friend and we couldn't find the answer given what we know, which is only fairly basic calculus, similar to what you would learn in the first year of 'A ...
0
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2answers
29 views

How to solve goniometric equation where function arguments differ?

I'm preparing myself for exams and I stumbled over goniometric equation: $$\cos^2x + \cos{2x}+1=0$$ Normally, those equations are solved by pretending that $\cos{x}$ is some variable $u$ and ending ...
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2answers
15 views

Graphs and Quadratic Equations?

Question: A ball is dropped from a height of $18$ metres. The quadratic equation $h = -2t^2 + 18$ gives you the height, $h$ metres, of the ball, after $t$ seconds. After how many seconds does the ...
2
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2answers
58 views

Show that $2\cosh z + \sinh z = i$

The equation is $$2\cosh z + \sinh z = i$$ I used the following formulas: $$\cosh z = \frac{e^z + e^{-z}}{2}, \sinh z = \frac{e^z - e^{-z}}{2}$$ to reduce this equation to $$3e^z - e^{-z} = 2i$$ but ...