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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-1
votes
1answer
11 views

Two circular tangents

The total area of both circles is $230$ $m^2$, i need to find the radius of each circle. The circles are externally tangential and the distance from their centers are $11$$m$. Unable to upload ...
0
votes
2answers
16 views

If a line parallel to $y=-7x+3$ touches the parabola $2x^2-3x+2$ in the point $(x_0,y_0)$, what is the value of $4x_0+y_0$?

I tried solving this but I've no idea how to find the point where a line of the form $y=-7x+n$ intersects a given parabola. Hints are welcome. Don't know calculus (don't even know if it's applicable ...
2
votes
3answers
33 views

The sum of the cubes of the reciprocal values of the roots of the equation $x^2+ax+1=0$ is?

$a$ is a real number. I tried solving this just by brute forcing it, but I get expressions that are ugly and pretty sure would yield nothing in the long run. So there is probably a trick to doing ...
0
votes
1answer
22 views

Intersection of two parabolas where one is vertex shifted

I would like to be able to calculate the intersections of two parabola's which accounts for one or both of the parabola's being shifted along the x axis I have written an excel vba function to do ...
1
vote
1answer
31 views

Steve Nash’s expected value from his one-and-one free throw situation is 1.72 points. What is his free-throw percentage?

The one-on-one free throw situation works like this - for the first throw, if you make it, you get to do it again. If you miss, you don't get another chance. If you make it the second time, you get ...
2
votes
0answers
26 views

Second-order Quadratic Constraint

I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1^T\theta\\ \text{subject to} & ...
0
votes
1answer
27 views

Estimation of Quadratic form parameters and convexity/concavity surface

I have 3 points in the 3-d space with their coordinates $(x~y~z)^T$. I would like to find the expression of the $\textbf{concave}$ quadratic surface that form those 3 points, i.e., $z=f(x,y)=ax^2 + ...
0
votes
2answers
44 views

I have no idea what below surface equation represent

I have the equation: $$x^2+y^2+4z^2-14xy+8xz-8yz=24$$ What does this equation represent? How can I find the "axes" of it (?), and is it possible to draw it when it intersect the plane $z=0$?
2
votes
1answer
59 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
votes
1answer
20 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. ...
0
votes
0answers
51 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 = ...
1
vote
2answers
47 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 ...
-1
votes
3answers
71 views

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

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?
-3
votes
1answer
53 views

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

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
votes
0answers
21 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
votes
1answer
17 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 ...
19
votes
5answers
300 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
votes
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
votes
2answers
51 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? ...
0
votes
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
votes
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
votes
4answers
190 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: $$ 2^{n-3}=\frac {20}{n} $$ ...
1
vote
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
votes
3answers
190 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 $-\sqrt{5}/5$
5
votes
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.
2
votes
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 ...
0
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
votes
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 ...
1
vote
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 ...
2
votes
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 ...
3
votes
2answers
66 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 ...
1
vote
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 ...
-6
votes
3answers
61 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?
0
votes
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 ...
5
votes
4answers
191 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 ...
1
vote
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
votes
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
votes
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
votes
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
votes
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
vote
1answer
61 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
votes
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
votes
3answers
610 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 = ...
10
votes
3answers
135 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
votes
0answers
18 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
vote
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 ...
1
vote
1answer
37 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 $ ...
2
votes
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 : ...
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 ...
3
votes
3answers
81 views

Prove $x^2 - x + 1 $ is always positive.

While solving a question, I came up with an inequality : $(1+x)(1-x+x^2)>0$ The book stated - where $(1-x+x^2)$ is always positive as $D<0$ and $a>0$ I'm not that sure how did it ...