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
28 views

Solutions of $\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$

$\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$ I get that $-3$ as a solution, but apparently 1 is as well a solution, and I don't see a mechanism through which I could find it. Any help would be ...
0
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1answer
18 views

Values of $p$ for which equation $p3^x+2\cdot 3^{-x}=1$ has a unique solution

$p3^x+2\cdot 3^{-x}=1$ I got this down to a quadratic equation by marking $3^x$ as $t$ and I fiddled with the stuff and got some solutions that apparently don't fit the real one in the textbook was. ...
1
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1answer
33 views

Dual plot for complex roots of quadratic equation

Real roots of quadratic equation $ x^2 - \sqrt 3 x + 1/2 =0 \tag{1} $ can be plotted on $x$- axis as its parabola intersection at $ (\sqrt 3/2 \pm 1/2,0). $ In an improvization I assign ...
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0answers
36 views

Confusion regarding dF/dx=0, F=constant

I thought i found a theorem "Given a curve in the (y,x) plane defined by DE $\frac{dy}{dx} = f(y(x),x)$ and if there exist a directional derivative of F along this curve satisfies relation $g = ...
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3answers
66 views

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then $a=b=c=0$ [on hold]

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then it is an identity i.e. it is true for all values of $x$ and $a=b=c=0$. What is a proof of this?
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1answer
49 views

Find the width of a rectangle with an area of $x^2 -4x -12$ and the length of $x-2$

There is a rectangle with an area of $x^2 -4x -12$. The length is $x-2$, what is the width? I'm having serious trouble solving this, can anyone help?
0
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0answers
19 views

Solving the quadratic optimization problem with quadratic inequality constraint

I have a quadratic optimization problem which which both objective function and constraint are convex. As the problem is very big, I used decomposition technique and divide the problem to smaller ones ...
0
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1answer
16 views

Finding Both Missing Co-ordinates in distance formula

Hi I am using this to find location of a device in a 2d plane based on the distance formula. The co-ordinates of reference points and the distance of the device from the device is known. How can we ...
3
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1answer
328 views

$y =f(x) =(ax^2 + bx +c)/(dx^2+ex+f)$ We have to find the conditions for this it takes all real values.

$$ y=f(x)=\frac{ax^2+bx+c}{dx^2+ex+f} $$ We have to find the conditions for this it takes all real values. MY solution One approach is to equate it to y and for a quadratic of x and put discriminant ...
0
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0answers
20 views

Find parameters of a quadratic surface given 3 points

I have 3 points in the space each defined as a vector with its two coordinates $\eta_k=(x~~ y)^T$. Given $\eta_1,~ \eta_2$ and $\eta_3$ I would like to find the parameters $Q,~ P$ and $b$ of the ...
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5answers
284 views

Probability of $ax^2 + bx + c = 0$ having real solutions

$a$, $b$, $c$ are random integer numbers between $1$ and $100$ (including $1$ and $100$, and uniformly distributed). What is the probability that the equation $ax^2 + bx + c = 0$ has real ...
5
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4answers
131 views

How to solve an equation with $x^4$?

Today, I had this question on a Maths test about Algebra. This was the equation I had to solve: $$(1-x)(x-5)^3=x-1$$ I worked away the brackets and subtracted $x-1$ from both sides and was left with ...
3
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2answers
50 views

Can anyone help me solve this?

Two taps A and B can fill a swimming pool in $3$ hours. If turned on alone, it takes tap A $5$ hours less than tap B to fill the same pool. How many hours does it take tap A to fill the pool? ...
132
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18answers
14k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
0
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1answer
55 views

Reverse Polish Notation Quadratic formula

The quadratic formula is $$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ I tried converting this to RPN; I am new to doing this, and I have thus: b-ac*4*-b2^+±a2*/. Am I ...
0
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1answer
37 views

Real world example of need for quadratic equation

I am (re)learning the quadratic equation. Having a concrete understanding of its purpose would really help, but I can not find any examples of a real-world scenario that requires the use of it that ...
3
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4answers
188 views

Solving equations with exponentials and a non-exponential term.

I know how to solve exponential equations. Just use logarithms, e.g., $$ 2^x-3=0 \\ 2^x=3 \\ x=log_23 \\ $$ But on a recent math test I found an equation of the form: $$ 2^{n-3}=\frac {20}{n} $$ ...
5
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3answers
54 views

Intuitive understanding of factoring quadratic equations [duplicate]

When factoring a second degree equation $ax^2 + bx + c$ you find the roots then take $a(x - \text{root})(x - \text{root})$. I am wondering why this works. Sorry if poorly phrased question.
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2answers
53 views

To find $x$ in $x^2 -8x-11=0$ [closed]

$x^2 -8x-11=0$ I have tried factorising but it won't factorise into a quadratic equation Hi, It would be great if you could complete this question with working and post it. Thx The two solutions of ...
4
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3answers
189 views

How to find cotangent?

Need to find a $3\cot(x+y)$ if $\tan(x)$ and $\tan(y)$ are the solutions of $x^2-3\sqrt{5}\,x +2 = 0$. I tried to solve this and got $3\sqrt{5}\cdot1/2$, but the answer is $-\sqrt{5}/5$
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2answers
185 views

Finding (sin(A+B))^2 given roots of a quadratic equation.

If tan A and tan B are the roots of the equation x^2 -ax + b = 0, then the value of sin(A+B)^2 is? Options are: ((a^2)/((a^2)+(1-b)^2), (a^2)/(a^2+b^2), a^2/(b+a)^2, a^2/(b^2*(1-a)^2) The value ...
2
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2answers
48 views

Finding conditions to make roots of a quadratic less than one in magnitude

I'm doing a problem that asks for you to find the conditions that make $y$ defined: $$y=x^2-bx+c$$ have real roots with magnitude less than one. Now the condition for the roots being real seems to ...
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1answer
40 views

How would find the general solution of the following differential equation [closed]

Could someone give me some help with finding the general solution of the below differential equation: \begin{equation*} \frac{ds}{dt} = \frac{(s^2 + st + t^2)}{t^2}\end{equation*} Thanks very much
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10answers
3k views

Why are equations written by equating something to zero?

A linear equation is $$ ax + b = 0 ; \,\, \,\, a\neq 0 $$ A quadratic equation is $$ax^2 + bx + c = 0 ; \,\, a\neq 0 $$ And so on... Why are all these equations written as $\dots = 0 $? Why do ...
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2answers
53 views

Let $y=x^2+ax+b$ cuts the coordinate axes at three distinct points. Show that the circle passing through these 3 points also passes through $(0,1)$.

Let $y=x^2+ax+b$ be a parabola that cuts the coordinate axes at three distinct points. Show that the circle passing through these three points also passes through $(0,1)$. Since, the graph of the ...
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votes
5answers
34 views

How to know if equation can be solved by factorising before trying?

So, I have core 1 test tomorrow and there is a lot of solving of quadratic equations without calculator and my weakest point is the time I waste in trying to factorise and equation but then it ends up ...
3
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3answers
36 views

Finding real coefficients of equation given that $a+ib$ is a root

Below is the question present in a past examination paper. I'll be giving my attempts and how I thought it through. Do feel free to point out any mistakes I make throughout my working even if ...
3
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2answers
65 views

Can you solve a quadratic equation using matrices?

I was wondering whether there are any alternatives or more efficient methods to finding a solution to a quadratic equation other than simply trial and error or by using the quadratic formula. I was ...
2
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2answers
62 views

Number of fingers of a Martian

I have a question about what seems to be modular arithmetic, but I can't quite get the answer. The problem goes along the lines of: It is often said Earthlings use the decimal system because they ...
1
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1answer
25 views

Irreducible quadratic “within” reducible quadratic

If we have a reducible quadratic function \begin{equation*} P(x)=a_1x^2+b_1x+c_1=(rx-x_1)(tx-x_2),~x_1,x_2,r,t\in\mathbb{Z}, \end{equation*} does there exist another irreducible quadratic function ...
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3answers
59 views

What is the solution of $a^2=b^2$? [closed]

How to solve $a^2=b^2$? Should I consider if the number is negative or positive?
1
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2answers
45 views

Find roots for an equation with quadratic, linear and log terms?

I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the form $ax^2 + bx + c\log x=0.$ considering the natural $\log$. Wolfram alpha is leading me to ...
5
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4answers
189 views

Solving an exponential equation involving e: $e^x-e^{-x}=\frac{3}{2}$

In a previous exam, my professor had the question \begin{equation*} e^x-e^{-x}=\frac{3}{2}. \end{equation*} I attempted to take the natural log of both side to solve it, but evidently that was ...
0
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1answer
27 views

Complex Coefficients and Real roots

Find $m$ which is a real number so that this equation has a real root. $2z^2-(3+8i)z-(m+4i)=0$ I've tried $b^2-4ac=0 $ but I can only seem to get complex $m$ values, so either I'm missing a key ...
1
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1answer
46 views

Solve for x without using the quadratic formula

Some context: I'm doing an inverse transformation method where I have the probability density function split in three parts. The first part is: $$ f_1:\frac{x-6}{8} $$ For $ 6 < x < 8 $. I ...
2
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1answer
31 views

Finding the maximum of sum of coefficients of a polynomial

Suppose $p(x)=ax^2+bx+c$ is a quadratic polynomial with real coefficients and $|p(x)| \leq 1$ for all values of $x$ in the range $[0,1]$. Prove that maximum possible value of $|a|+|b|+|c|$ is $17$. ...
0
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3answers
31 views

If a quadratic equation can have less then two solutions

is there anyway that a quadratic equation has less than two solutions? If the first coefficient a is 0, then it is not a quadratic.
0
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0answers
10 views

Solution to Equation involving Volatility

The following question will have little context, though, it is not relevant. To summarise though, I am trying to find solutions $u$ and $d$ to the following equation given that $d = \frac{1}{u}$: ...
3
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3answers
1k views

How to deduce $\,n^2+5n-12=0\,\Rightarrow\, n^3 = 37n - 60$?

Given n is a root of quadratic equation $x^2+5x-12=0$. Show that $n^3=37n-60$. Does this question have any trick or require any special mathematical skill?
1
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1answer
59 views

how to work out 3 equations simultaneously

So i was doing this linear programming question and got stuck on this part, so how do you workout simultaneously $2x + 3y = 30 $ $(2/3)x + 2y = 16 $ $(16/3)x + 4y = 64$ According to lpsolve we ...
2
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1answer
23 views

Interval of $a$ for which the solutions of the equation $x^2-6ax+2-2a+9a^2$ are bigger than $3$

The question is asked so that I have multiple choices and need to prove the thing both ways (it's an equivalence). The problem is, whichever thing I compare to the minimal values of the roots I try ...
7
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3answers
609 views

Universal quadratic formula?

Is there any way to write the quadratic formula such that it works for $ac= 0$ without having to make it piecewise? The traditional solution of $x = (-b \pm \sqrt{b^2 - 4ac}) / 2a$ breaks when $a = ...
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3answers
130 views

What is the connection between the discriminant of a quadratic and the distance formula?

The $x$-coordinate of the center of a parabola $ax^2 + bx + c$ is $$-\frac{b}{2a}$$ If we look at the quadratic formula $$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ we can see that it specifies two ...
2
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0answers
17 views

Extraction of quadratic terms with state-space representation

I am having trouble with transforming the dynamics of a 4DOF gyroscope to a neat state-space representation. The system has the following set of equations: $T_i + f_i(\omega, \alpha) = 0;\;i:1-4$ . ...
1
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1answer
32 views

Factor polynomial with irrational roots using quadratic equation

If I want to factor the polynomial $x^2 + 3x + 1$, I thought I could use the quadratic formula to find that its roots are $\dfrac{-3\pm\sqrt{5}}{2}$. Then, since those are both negative values, take ...
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4answers
184 views

Roots of $x^2+3x+2=0$ are infinite !!! [closed]

I have quadratic equation here: $x^2+3x+2=0$ so $(x+2)(x+1)=0$ and I can do $(x+2)=0/(x+1)$ and that solution of the equation is $x+2=0$ so $x=-2$ but my teacher said that it is wrong why? ...
2
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3answers
42 views

For what real values of $a$ does the range of $f(x)$ contains the interval $[0,1]$?

Question : For what real values of $a$ does the range of $f(x) = \cfrac{x+1}{a+x^2} $ contains the interval $[0,1]$? My doubt lies in the further preceding of this question. The book states : ...
1
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1answer
36 views

Difference and Quotient of roots of a quadratic equation

In school we are taught the sum and product of roots of $y= ax^2+bx+c$. But are not the difference and quotient of roots equally important? Difference $= \dfrac{\sqrt{b^2-4ac}}{a}$ and Quotient $ ...
3
votes
3answers
54 views

Find the equation whose roots are each six more than the roots of $x^2 + 8x - 1 = 0$

Find the equation whose roots are each six more than the roots of $x^2 + 8x - 1 = 0$ I must use Vieta's formulas in my solution since that is the lesson we are covering with our teacher. My ...
1
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3answers
78 views

Convert quadratic bezier curve to parabola

A quadratic Bézier curve is a segment of a parabola. If the $3$ control points and the quadratic Bézier curve are known, how do you calculate the equation of the parabola (which is an $y=f(x)$ ...