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Questions tagged [quadratics]

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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1answer
10 views

The greatest area for a rectangle on a track field.

An athletic field with a perimeter of 0.25 miles consists of a rectangle with a semicircle at each end, as shown below. Find the dimensions that yield the greatest possible area for the rectangular ...
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1answer
28 views

Quadratic Functions - vertex of graph proof

Can you explain how $x$ becomes $x + b/2a$ and $c$ becomes $4ac - b^2/4a$ all of sudden? Can you please explain at a Pre-Calculus level, thank you very much.
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0answers
20 views

Looking for two quadratic equations that intersect at (0,0) and (1,1) [on hold]

Can someone please provide me with two quadratic equations that intersect at (0,0) and (1,1). I'm also going to need 100 points along each line from (0,0) to (1,1) and I'm not sure how to obtain these....
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1answer
14 views

Discriminant when graph lies above or below the x axis.

Suppose a quadratic equation has been given where the a value (ax^2 + bx + c) is a positive and it has been said that the graph of the equation lies above the x-axis- what is the discriminant? For ...
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0answers
18 views

How to find the least common multiple with an acceleration component

Is it possible to find the least common multiple of two numbers when there is an acceleration or deceleration component. For example if I have two parabola's which intersect at a known given positive ...
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1answer
18 views

How to know if a graph is exponential just by looking at the data values

If I were given the points (1,3) (2,5) (3,7) and assumed this pattern continued forever, I know that it is linear as there is a constant the y value for an increase of one for the x value. If I were ...
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0answers
12 views

Intervals of a Multivariable Function

If the gradient at some point of a multivariable function equals $\vec{0}$, and the Hessian is positive or negative semidefinite, is there a notion, as in single variable calculus, of resolving the ...
3
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1answer
119 views

Finding the minimum value.

I'm struck on this question, I tried hard but couldn't solve it. Question: if a quadratic equation in $x$: $$ax^2 - bx + 5 = 0$$ does not have two distinct real roots, then find the minimum value of $...
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0answers
32 views

Find the roots of a quadratic [on hold]

1) Show that: $(a^2+1)(β^2+1)=(c−1)^2+b^2$ 2) Find In terms of b and c, a quadratic whose roots are $\frac a{a^2+1}$ and $\frac β{β^2+1}$ I do not know how to employ the first information in 1) to ...
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1answer
16 views

If the parabola is translated from its initial position to a new position by moving its vertex along the line $y=x+4,$

The parabola $y=4-x^2$ has vertex $P.$It intersects $x-$axis at $A$ and $B.$ If the parabola is translated from its initial position to a new position by moving its vertex along the line $y=x+4,$ so ...
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3answers
46 views

Find all $a, b \in \mathbb R$, ($b\ne0)$, such that the roots of $x^2+ax+a=b$ and $x^2+ax+a=-b$ are 4 consecutive numbers

Find all $a, b \in \Bbb R$, ($b\ne0)$, such that the roots of $$x^2+ax+a=b$$ $$x^2+ax+a=-b$$ are 4 consecutive numbers. We have: $$x^2+ax+a-b=0$$ $$x^2+ax+a+b=0$$ $x_1, x_2$ - roots of first equation;...
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4answers
37 views

Having issues understanding fraction division when applying quadratic formula

I'm trying to apply the quadratic formula, and having trouble understanding how: $$\frac{-3 ± 3\sqrt{41} }{-18}$$ evaluates to $$\frac{-1 ±\sqrt{41}}{-6}$$ and not $$\frac{1}{6}±\frac{\sqrt{41}}{...
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4answers
35 views

When I complete the square on $3x^2 - 12x + 14$ I get an imaginary number, where have I gone wrong?

I have a question in my excersise book: By completing the square show that the expression $3x^2 - 12x + 14$ is positive for all $x$ My approach was to complete the square and rearrange to make $x$ ...
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3answers
58 views

How do you turn an irrational, non-transcendental number, like 1.618… back to its form of (a + sqrt(b))/c.

Looking at irrational numbers, I had an idea, as to computing square roots. Take the golden ratio. Numerically, it's 1.618.... but I can also write it like this: $\frac{1+ \sqrt{5}}{2}$ I want to ...
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2answers
29 views

How to factorize $zz^*-4z-4z^*+12=0$ (where $z^*$ is the complex conjugate of $z$)

I'm trying to factorize this: $$zz^*-4z-4z^*+12=0$$ to get this: $$|z-4|^2 - 4 = 0$$ where $z=x+yi$ is a complex number and $z^*=x-yi$ is the conjugate complex number of $z$. I'm trying to factorise ...
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1answer
41 views

Piece-wise quadratic function [closed]

How can I find a minimum of a piece-wise quadratic function? (minorant of a set of quadratic functions) An example of this will be appreciated.
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5answers
57 views

Information lost in solving system of quadratic equations

I have a system of two quadratic equations $$ \left\{ \begin{array}{c} 2x^2+x-1=0 \\ 2x^2+5x+2=0 \end{array} \right. $$ I tried to solve it the following way: $$ 2x^2=-5x-2$$ substituting in the ...
1
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1answer
27 views

The Shading of Double Petaled Flowers

Here is the shape that I am trying to shade in. I have the outline. Here are the equations that I used: I was wondering how I could manipulate these domains and ranges or maybe the equations. I was ...
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3answers
35 views

Find the equation of a line intersecting a parabola

Okay here's the question: Consider the parabola P of equation $y=x^2$, and the line $L$ of equation $y=x+6$. Let $P(x_p,y_p)$ be a point on the arc of the parabola P below L. Let A and B be the ...
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1answer
9 views

Quadratic Yield Response Function [closed]

How do I find the quadratic yield response function in the form of "Y = b0 + b1*X + b2*X^2" for a set of data in excel?
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0answers
44 views

Show that this system has at least one unbounded solution as $t \to \infty$

Assume the system $$x'(t)=\begin{pmatrix} \frac12-\cos t & 2 \\ 1 & \frac32+\sin t \end{pmatrix}x(t)=A(t)\cdot x(t)$$ with minimum period: $T=2\pi$. Let $\mu_1,\mu_2$ be its characteristic ...
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2answers
45 views

Show that $|\frac{1}{2n}-\frac{1}{2m}| < \epsilon$ holds for all $m, n > \frac{1}{\epsilon}.$

In Example 1.5-9 of the book Functional Analysis by Kreyszig it claims that $|\frac{1}{2n}-\frac{1}{2m}| < \epsilon$ holds for all $m, n > \frac{1}{\epsilon}.$ My calculations don't lead to ...
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6answers
28 views

How would one factorise $m^2 + (2AB)m + B^2 =0$

How would one factorise $m^2 + (2AB)m + B^2 =0$, to go onto solve a second order differential equation
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1answer
40 views

How to graphically depict the possible solutions of a quadratic equation

I have the following quadratic equation : $$am^2 + bm + (c_1^2 +c_2^2) =0,$$ where the solution is given by $$m = \frac{-b\pm\sqrt{b^2-4a(c_1^2+c_2^2)}}{2a}.$$ Here, $\Delta>0$. Thus I have ...
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1answer
27 views

Is there a formal name for an equation with multiple solutions?

I saw that there is a related question for an equation with no solution, but I was curious about an equation with more than one solution.
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1answer
39 views

Root of an quadratic equation

I have the following quadratic equation : $m^2 + m(p-1/l) - (\Omega_x^2 + \Omega_y^2)=0$ I would like to get the solution in terms of $\Omega_x, \Omega_y$ with some approximations i.e. neglecting $(...
6
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5answers
658 views

Why is my solution incorrect for solving these quadratic equations?

$$\frac2x -\frac5{\sqrt{x}}=1 \qquad \qquad 10)\ \frac3n -\frac7{\sqrt{n}} -6=0$$ I have these two problems. For the first one I create a dummy variable, $y = \sqrt x$ then $y^2 = x$. Substituting ...
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0answers
7 views

How to linearize the product of a binary variable X and a continuous variable Y

I have equations and I want to linearization them they are c>=(st+pr)x where st is continuous variable , pr is continuous variable and x is binary variable. c=>st(x+z) where st is continuous ...
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3answers
22 views

How can I find the equation of one parabola given the equation of another parabola?

The equation of the other parabola has to follow the form: $4p(x – h) = (y – k)^2$ because it is a sideways parabola. I can see that the vertex is at (-4.5,18) So then the equation would be $4p(x+4....
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3answers
44 views

Solve quadratic equation by completing square

We have a equation: $2x^2+7x+3$ I tried to find the vertex of the parabola by this formula: $a(x-h)^2+k$ but I could not get it. I got this: but it is not right. $2(x^2+ \frac{7}{2}x+\frac{49}{4})+3-\...
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3answers
87 views

For a given quadratic function, identify roots, vertex coordinates, and axis of symmetry [closed]

For each graph of quadratic functions shown, identify A) $x^2-3x-4$ B) $2x^2+7x+3$ 1) the solutions, or roots 2) coordinates of the vertex 3) equation of axis of symmetry
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0answers
16 views

What are other “complicated” examples of factoring with dummy variables?

I'm looking for many examples of questions that are like these: I want to specifically make use of the strategy where I create dummy variables
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3answers
36 views

Quadratic questions involving 3 variables [closed]

If $(x-2)^2+(y-3)^2+(z+4)^2=0$ then what is the value of $x+y+z $.
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4answers
39 views

Condition in terms of b and a if $ax^2+bx+c=0$ has two consecutive odd positive integers as roots

The roots of the equation$$ax^2+bx+c=0$$, where $a \geq 0$, are two consecutive odd positive integers, then (A) $|b|\leq 4a$ (B) $|b|\geq 4a$ (C) $|b|=2a$ (D) None of these My attempt Let p and ...
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1answer
36 views

How do I solve this quadratic-intersection question?

My question: Find the values of $k$ for which the parabola $y=2x^2+kx+9$ does not intersect the line $y=2x+2$. My workings: I am thinking of using the discriminant rule to this where Δ < 0, ...
2
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1answer
33 views

Quadratic Drawing

I was wondering if anyone could help me with the shading This is the picture: Here are my equations so far: I was wondering if anyone knew what domain and range constraints I could add to make only ...
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4answers
44 views

Find the range of values which has no real solutions

I would like to know how to solve the following problem: Find the range of values of the parameter $m$ for which the equation $2x^2 - mx + m = 0$ has no real solutions. I know I have to use the ...
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2answers
33 views

How do you solve these 2 equations?

$$xy = 1/6$$ $$y+x = 5xy$$ I tried solving them using all methods - substitution, elimination and graphing - but can't get the solutions
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1answer
33 views

Solving this system equations?

I have to solve this system of equations with $(x,y,z) ∈ ℝ$ $x^2 + y + z = q$ $x+ y^2 + z = q$ $x + y + z^2 = q$ for $q = -1$ So we have: $x^2 + y + z = -1$ (1) $x+ y^2 + z = -1$ (2) $x + y +...
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3answers
28 views

Equation solving?

I've got this system of equations. (a,b,c) ∈ ℤ $a*b+1 = c$ $a^2 + b^2 +1 = 2c$ $2a + b = c$ I tried to substitute a little bit: $a^2 + b^2 + 1 -(2a+b) - (a*b+1) = 0$ Ultimately: $a(a-2-b) + b(...
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2answers
83 views

If $x^2-5x+1=0$ what is value of $\frac{x^{10}+1}{x^5}$

If $x^2-5x+1=0$ what is value of $\frac{x^{10}+1}{x^5}$. I tried with calculator but I don't think that,that was proper method ,if you got one pls post it
3
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4answers
212 views

Proof without words of the Quadratic Formula?

As suggested by @Moti and @YvesDaoust in this post, a simple way to identify the roots (red dots) of a parabola (given focus and directrix, blue) by means of straightedge and compass is to draw the ...
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6answers
41 views

Where is the logical flaw in solving this equation?

I ran across this equation... $\sqrt {2x+6}+4=x+3$ Without thinking, I solved for x in the following way: $\sqrt {2x+6}+4=x+3$ Subtract 4 from both sides. $\sqrt {2x+6}=x-1$ Square each side. ...
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2answers
95 views

Approximation of the quadratic formula with straightedge and compass

Given a directrix and a focus (blue), we can define a parabola as illustrated below. We suppose the parabola intersecting the $x$-axis in correspondence of the red dots. We draw the line ...
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1answer
125 views

Solve the system of $3$ quadratic equations [closed]

Consider the system of equation $${{x}^{2}}+{{(1-y)}^{2}}=a\\ {{y}^{2}}+{{(1-z)}^{2}}=b\\ {{z}^{2}}+{{(1-x)}^{2}}=c$$ Compute $x(1-x)$ in terms of $a,b,c$.
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0answers
35 views

Max no of real roots

$p(x) =x^6 + ax^5+ bx^4 +x^3 + bx^2 + ax + 1$ Given that p(1)=0 but p(-1) is not zero. What is maximum number of distinct real roots of p(x)? I divided by $x^3$ and made a cubic in t, where $t=x+1/x$...
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1answer
19 views

Formula Rearrangement

Hi StackExchange community, The formula has this form: $$ {-7 \pm X \over \sqrt{2} - 3}-3.$$ How can I rewrite this to be more compact ? Thank you.
3
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1answer
58 views

How to find x in $ax^2+bx+c=0$ [duplicate]

Quadratic equation, $$ax^2+bx+c=0\tag1$$, then the answer to x is $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\tag2$$ How did they get from steps $(1)$ to $(2)$? This look impossibe!!
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1answer
34 views

Solving a quadratic formula with positive discriminant yields only one correct solution.

I'm a math tutor at a small university. One of my students asked me about the problem, $$p - 2\sqrt{p} = 15$$ Solving this, we found, in sequence, $$-2\sqrt{p}=15 - p$$ $$4p = p^2 - 30 p + 225$$ $$p^2 ...
1
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2answers
77 views

Find the value of $a$ such that $f(x)$ has exactly one root $\alpha$ in interval $(1,2)$ and…

Question Find the value of $a$ such that equation $$f(x)=x^2+(a-3)x+a=0$$ has exactly one root $\alpha$ between the interval $(1,2)$ and $f(x+\alpha)=0$ has exactly one root between the interval $(0,...