Questions on quadratic equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-h)^2+k$ or $y=a(bx+c)(dx+e)$.

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Quadratic Functions Word Problem

A holding pen is being built alongside a long building. The pen requires only three fenced sides, with the building forming the fourth side. There is enough material for 90m of fencing. Predict what ...
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solve an quadratic equation

I was reading a document , where I stucked in figuring out this equation. $f(k)= k^2-nk+\frac{n^2 - n}{2}$. This is a quadratic function of $k$. It is minimized when $k=\frac{n}{2}$ (the $k$ ...
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2answers
86 views

Find the value of $f(x)$ for $x = 2 + 2^{2/3} + 2^{1/3}$

If $x = 2 + 2^{2/3} + 2^{1/3}$, then find the value of $f(x)=x^3 - 6x^2 + 6x$. I am unable to get to the answer - end up with more than one term. Please help me solve this!
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132 views

For the given quadratic equation find the value of p

For the equation 3x^2 + px + 3 = 0 , p>0, if one of the roots is square of the other, then p is equal to? Solving the equation, i get the value of p as -6 but the question states that p>0. Is there ...
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Quadratic Function: X intercepts.

A quadratic function with a y-intercept of 0 and an axis of symmetry of x=-1. Apparently, there is suppose to be 2 x-intercepts, which I really don't understand. How can the parabola cross the x ...
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Quadratic Functions word problem.

I have the quadratic function $$y=\frac{1}{294}(x-84)^2-24$$ that represents the shape of the cables of a certain bridge. I am suppose to be determining the vertical height of the cables above the ...
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72 views

What is the spherical parametrization of an ellipsoid NOT centered in the origin?

I would like to know how to parametrize an ellipsoid not centered in the origin, but with its axes parallel to the main axes of the reference system. The result I am looking for would be an ...
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Quadratic Functions Problem

Question: The main section of a certain bridge has cables in the shape of a parabola. Suppose that the points on the tops of the towers where the cables are attached are 168m apart and 24 vertically ...
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For what real number $c$, this equation has exactly three solutions?

For what real number c does the equation $|x^2 + 12x + 34| = c$ has exactly three solutions?
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105 views

Finding dimensions using quadratic formula

A 52 m long fence is constructed on three sides of a housing block with area 240 m^2. The fourth side facing the road is left open. Find the dimensions of the block. Also here's another question I ...
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Problem Solving quadratics

A rectangular paddock has perimeter of 600 m and area 21 600 m^2. Find the dimensions of the paddock. So far, I've figured out the formula is x(300-x)=21600 and rearranged to 300x-x^2=21600. I'm not ...
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Quadratics: Solve for B and C.

If I know that the points (-1,6) and (2,3) are on the graph of the quadratic function f(x) = 2x^2 + bx + c, how do I determine b and c? Thanks to anyone who helps.
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Quadratic Functions: Determine the value of b

I'm having trouble with this question and I'm not sure what to do. Would appreciate any one who helps me out. Question: The point $(-2,1)$ is on the graph of the quadratic function: $f(x) = -x^2 + bx ...
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2answers
110 views

Quadratic Formula for complex variable with real coefficients

I have been upto proving the following $$(\forall x\in \mathbb C, ax^2 + b x + c = 0) \land(a\neq0)\Leftrightarrow (x = {\frac {-b \pm \sqrt{b^{2}-4ac}} {2a}})$$ Due to equality we need to proof bi ...
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73 views

Finding x using the pythagoras theorem

$$x^2 = (x+1)^2 + (x-7)^2$$ can someone please find $x$? Also this is a quadratic equation problem solving question.
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46 views

Consider a quadratic equation

$x^2-2(a+1)x+3a+2=0$ Where a is any real number. Find all values of a so that the equation has two distinct real roots. I tried solving for a by using the inequality $b^2-4ac\gt0$... but when i ...
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what would be the way to solve a system of equations like this one?

Solve: $xy=-30$ $x+y=13$ {15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?
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Quadratic problem

NASA launches a rocket at $t = 0$ seconds. Its height, in meters above sea-level, as a function of time is given by $h(t) = -4.9t^2 + 37 t + 101$. Assuming that the rocket will splash down into the ...
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Equation with the discriminant

I'm stuck on my maths homework, and would appreciate help. The question is: Show that if the equation $(m + n)x^2 - 2mnx - (m-n) = 0$ has equal roots, then $$m^2 = \frac{n^2}{1-n^2}$$ I've worked ...
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58 views

What is the relationship between three points on a quadratic curve and the curves coefficients?

In other words, is there a formula to get the coefficients a,b and c in terms of three points $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3, y_3)$? I am asking this because I have a linear algebra problem that ...
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Question on simple quadratic word problem regarding weekly revenue when price of merchandise is lowered

A store owner sells headphones at 24 dollars a piece with roughly 1000 sold per week. The store owner finds that for every 1 dollar decrease on the price per headphone he sells 100 more headphones per ...
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Quadratic Equation Algorithm

I need help for developing an algorithm. I'm drawing a given amount of line segments and I want them to decrese in length the more they get close to amount/2 and then to grow again until they reach ...
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45 views

Can one solve this system of qudratic equations unambigiously?

Given the parameters $p_1, p_2, p_3$ I want to know if the following system can be solved: $p_1 a + p_2 c + ef = 0\\ p_1 b + cd + p_3 f = 0\\ ab + p_2 d + p_3 f = 0\\ p_1^2 +a^2+b^2=1\\ ...
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If the equation $|x^2+bx+c|=k$ has four real roots …

If the equation $|x^2+bx+c|=k$ has four real roots then which of the following option is correct : (a) $b^2-4c >0$ and $0<k<\frac{4c-b^2}{4}$ (b) $b^2-4c <0$ and ...
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Tangent of a rational function

I'm wondering if someone could help me solve a little conundrum I've been having. I'm trying to find the equation of a line with slope $-1$ tangent to the curve of a rational function. The function is ...
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Taking Calculus in a few days and I still don't know how to factorize quadratics

Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ...
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1answer
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Quadratic Equation Inequality

What is relation between a and b in below equations (I and II) - I. $9a^2+18a+5=0$ II. $2b^2+13b+20=0$ Options - ...
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Quadratic Prime

We had received some questions on Quadratic equations, But I wasnt able to do one. Here it goes: Let $a,b$ be natural numbers $a>1$. Also, $p$ is a prime number. If $ax^2+bx+c=p$ for 2 distinct ...
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118 views

First degree or second degree quadratic equation?

$X-\sqrt{121}= 0$ $Y^2-121=0$ When I try to solve by squaring on both sides the first equation changes to $X^2-121 =0$, where $x = \pm 11$. When I try to solve it this way: $x= \sqrt{121} = +11$, ...
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How to Find The Roots of this Quadratic Given Sum & Product

My question is: The sum of the roots of a quadratic is $55/72$, and the products of the roots is $-25/12$. Find the roots. How I'm trying to do it so far: (Also, please correct my thought process if ...
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Solve the equation $x^2+bx+c+t\log(x)=0$

There is a explicit formula to solve the equation $x^2+bx+c+t\log(x)=0$ with the constraint $x>0$?
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5answers
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Algebraic process to find numbers so that $xy=45$ and $x+y=18$

Can someone help me with the following question? The sum of two numbers is $18$ and their product is $45$. Find the numbers. I know that the answer is $15$ and $3$. But how do I find that answer ...
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1answer
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Quadratic equations - finding value of x

Question: Solve the equation $$\frac{2x}{3x^2 - x+2} - \frac{7x}{3x^2 + 5x+2} = 1$$ I attempted to split the two quadratic equations into their roots, however, was unable to do so. Then I tried to ...
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4answers
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Quadratic equation - solving for x

Question: Solve the equation $$(x+2)(x+3)(x+8)(x+12) = 4x^2$$ I tried to solve the equation by expanding the LHS and then equating it to the RHS, but that just doesn't seem to be feasible. I am ...
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Problem involving a quadratic

This is the equation : $mx^2-(4m-1)x+3m-2=0 $ We are asked to find a relationship between the roots that doesn't involve the 'm' parameter. Therefore, I thought the key to achieving this was making ...
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How to solve the quadratic equation $x^2-1=2$?

Solve $x^2-1=2$ I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.
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How do I factor this quadratic?

I'm going through the AoPS Algebra book, and I'm on the quadratics section. I'm given this challenge question: $ \displaystyle 2x^2 + 7x(\sqrt{3}) + 9 = 0$ And I have to solve for ...
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2answers
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Using factoring to solve the equation $(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$

Solve for all values of $r$: $$(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$$ I'm not sure how my thinking isn't really correct here. I know this all seems very elementary and such, ...
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48 views

Quadratic equations ..

The set of non-zero values of k such that the equation $|x^2-10x+9| =kx$ is satisfied by atleast one and atmost three values of x, lies in _. The answer is $(-\infty, -16] \cup [4 , \infty) $. How ...
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Quadratic formula and factoring are leading to different answers

$$x^{ 2 }-2x-15=0$$ By factoring, I get: $$(x-5)(x+3)$$ Which has the solutions: $$x=5, x=-3$$ However when I use the quadratic formula (which is what the book saids to use), I get $$\frac { 2 ...
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1answer
68 views

How to solve the system of equations $y= 5x^2-2x$ and $y=10x+9 $?

I am trying to solve this system of equations $y= 5x^2-2x$ and $y=10x+9$. I have worked the problem out and I am lost could someone please explain it? I have a test tomorrow over this information.
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4answers
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Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
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Quadratic equation not equal to zero (solving a matrix with a parameter)

I came across this in my matrix module, learning about number of solutions when the matrix has parameters. $$ \left[ \begin{array}{ccc|c} 1 & ...
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Finding the points where a circle intersects an axis

A circle has the equation: x²+y²+4x-2y-11 = 0 What would be the coordinates of the points where the circle intersects with the y-axis and how would you calculate it?
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How to solve this word problem on the topic of quadratic equations?

The maths teacher of Mumbai is transferred to another school. The students of Class 10 decided to buy a book for 360 rupees(currency) as a gift for her. On the farewell day, 4 students did not turn up ...
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42 views

Difference between Half Quadratic vs Quadratic

Half quadratic minimization/penalty/optimization, I am unable to find any related material/resources. If anyone can point to some useful resources, it will be great
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Sequences and series

If $p, q, r$ are in G.P. and the equations: $$px^2 + 2qx + r = 0$$ $$dx^2 + 2ex + f = 0$$ Have a common root, then show that $$\frac{d}{p}, \frac{e}{q}, \frac{f}r$$ are in A.P. Well I tried taking ...
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Check if a point is within a quadratic surface (with arbitrary rotation)

Is there a general way to check whether a point is on a quadratic surface given that the principal axes do not need to coincide with the coordinate axes and that the quadric's centroid does not need ...
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28 views

Quadratic Equations and factorized form

I have trouble understanding why: Given the equation $x^2+4x-21=0$, the solution given is $(x+7)(x-3)=0$ in its factorization form. Using quadratic equations, $$ x = \frac{-b\pm \sqrt{b^2-4ac}}{2a} ...
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Can I perform the quadratic formula on polynomial with complex coefficient?

2 weeks ago, we had a Math test on complex number. One of the question was: Let $z=x+iy$ be a non-zero complex number, where $x,y \in \mathbb{R}$. Given that $z+\frac{1}{z} = k$, where $k$ ...