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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10
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3answers
118 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
11 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
26 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
34 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
40 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
45 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
77 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 ...
1
vote
3answers
59 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 $y=f(x)$ ...
0
votes
1answer
75 views

How do i expand/simplify this quadratic (or quartic?) equation

I'm having trouble doing the following question, was wondering if anyone was able to lend a hand, would be greatly appreciated as i'm not too sure where to start or how to go about this. The ...
0
votes
2answers
64 views

Approaching this proof problem? If $0 \le x \le 3$ then $12 - 7x + x^2 \ge 0.$

Prove that if $x$ is a real number in the range $12 - 7x + x^2 \ge 0.$ Which type of proof should I use to solve this? At first I thought direct proof. Choosing a number between $0$ and $3$ and ...
-1
votes
1answer
42 views

Quadratic equation using Pythagorean theorem [closed]

The width of a rectangular TV screen is 22.9 in longer than the height. If the diagonal is 60 in find dimensions of the screen. Using Pythagorean theorem express the given information in the ...
0
votes
1answer
21 views

Roots of the quadratics $x^2+5x+4$ and $x^2+8x-3$ modulo $45$ [closed]

how many roots of the two polynomials $x^2+5x+4$ and $x^2+8x-3$ in $Z_{45}$ have. Is there an easier way to do this ?
3
votes
2answers
78 views

Equality of a quadratic function

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ an arbitrary function and $g: \mathbb{R}\rightarrow \mathbb{R} $ a quadratic function with the following property: For any $m$ and $n$ the equation ...
1
vote
1answer
25 views

Trigonometric Equation, quadratic using two functions

I am struggling to know how to solve this equation as it involves more than one type of trigonometric function, I know how to do it with one repeated function. If a solution could be explained, that ...
0
votes
2answers
23 views

Quadratic using the roots of unity, where $\omega^7 = 1, \omega \neq 1$

Say that $\omega$ is a complex number, where $\omega^7 = 1, \omega \neq 1$. Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. $\alpha$ and $\beta$ are roots ...
0
votes
3answers
40 views

When can I divide both sides of an equation if one side is zero

Where K is some positive Integer For the following examples: $$ K(a+b)(p+q)=0 $$ $$ Ka^2+Kbx+Kc=0 $$ Can I just divide both sides of the equation by K (dividing into 0 on the right) and effectively ...
2
votes
0answers
51 views

Does $2^2=4$ imply $2=\pm \sqrt{4}$?

I read the square root property from the book, College Algebra by Raymond A Barnett and Micheal R Ziegler that, The square root property says, If $A^2=C$ then $A=\pm \sqrt{C}$ I took the ...
4
votes
4answers
55 views

Find all integers n such that the quadratic $5x^2 + nx – 13$ can be expressed as the product of two linear factors with integer coefficients.

I am unsure of how to approach this problem. I have thought about using the Rational root theorem, but I am unsure if this answers the question being asked. Using the theorem, I get $\frac{p}{q} = ...
0
votes
1answer
25 views

Interval in which roots lie.

For a quadratic equation, we have several conditions from which we can determine the interval in which the roots lie. eg: If exactly one of the roots of a general quadratic equation lies in the ...
1
vote
1answer
24 views

Quadratic function with positive integral coefficients problem

Here is the problem statement: Let $f(x)$ is a quadaratic expression with positive integral coefficients such that for every $\alpha, \beta\; \epsilon\; \Re$, $\beta>\alpha$, $\int_\alpha^\beta ...
0
votes
2answers
31 views

Roots of quadratic equation

If the roots of $ax^2+bx+c$ are $\alpha$ and $\beta$, express $\frac1\alpha-\frac1\beta$ in terms of $a$, $b$ and $c$. I know how to express $\alpha+\beta$ or $\alpha\beta$ which is usually enough, ...
0
votes
2answers
25 views

For which $m$ does the equation $\lvert x^2+5x+4 \rvert +x-m=0$ have more than two real solutions?

When dealing with problems containing quadratic equations, I've never come across one that deals with a number of roots that is bigger than 2. I've tried breaking up the function into two parts. ...
2
votes
4answers
51 views

Find the sum of the roots of a quadratic function given the vertex of its graph

Question: At this parabola $$y = ax^2 + bx - c$$ and vertex is $T(3,9)$. What is the sum of roots of this parabola ? Help or give a hint. Thanks
0
votes
1answer
31 views

Dimensions of a paddock (3 sides of a rectangle) to enclose maximum possible area

I need help with Qs 4, 5 and 6!! Three sides of a rectangular paddock are to be fenced, the fourth side being an existing straight water drain. If 1000m of fencing is available, what dimensions ...
2
votes
4answers
37 views

Simultaneous Quadratic Equations: $x^2 + y ^ 2 - 2 x + 6y - 35 = 0$ and $2x + 3y = 5$

I've been given the task to simultaneously solve: $$x^2 + y ^ 2 - 2 x + 6y - 35 = 0$$ $$2x + 3y = 5$$ I've tried applying the substitution method by reordering the second equation to both $x$ and ...
0
votes
2answers
11 views

Finding the set of all $k\in R$ so that the roots of equation $x^2-(5k+3)x+(k+3)^2$ obey $x_1<4<x_2$

Stuck on this one. I get inequations with square roots in them, ones which we aren't supposed to know to solve this problem.
1
vote
2answers
23 views

The equation $x^2+px-p$ has two real and different solutions $x_1$ and $x_2$ for which $1<(x_1/x_2+x_2/x_1)<3$ for which interval of $p$?

This is a multiple choice question, but I figure it will be easy enough to do without the given answers. It seems like an easy question. Use Vieta's formula from the inequality and define $p$ from ...
0
votes
1answer
35 views

The first assumption leads to the third one that looks inconsistent at a glance. Can you explain it better?

Background I am trying to solve the following problem: > Given 2 distinct curves $C_1: y=f(x)=e^{6x}$ and $C_2: y=g(x)=ax^2$ where $a>0$. The objective is to find the range of $a$ such that ...
0
votes
0answers
18 views

Line intersection with Sphere

I'm trying to get a formula for calculating intersection points of a line with a sphere (3d space). I've been following this one: Wiki Line-sphere intersection But I'm 99% sure that this one is ...
3
votes
2answers
161 views

Simple maths and a typo: impossible answer?

Long story short: While doing some simple math exercises, I came across one that seemed impossible. Days later I decided to search the web for it, and found out there was a typo, putting the $^2$ ...
1
vote
1answer
25 views

Difficulty turning a quadratic equation to “vertex”-form

I'm having difficulty reducing a quadratic equation to its "vertex-form" by following my textbook and nearly every tutorial I can find online. The starting equation is: $$f(x) = -2x^2 + 16x - 24$$ ...
0
votes
2answers
67 views

Online service for completing the square

Is there any online service for completing the square? For example:
1
vote
6answers
63 views

If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then find value of m.

Problem : If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then find value of $m$. My Approach: $|x^2+4x+3|-mx+2m=0$ Case I : $x^2+4x+3-mx+2m=0$ $\Rightarrow x^2+ x (4-m) + ...
0
votes
0answers
33 views

The property of roots of quadratic equation

I have a problem with two tasks: Given a quadratic equation $ax ^ 2 + bx + c = 0 $ (roots can be complex or real), $a, b, c \in Q$. Prove that ${x_1} ^ m + {x_2} ^ n \in Q$. We have a trinomial ...
0
votes
4answers
48 views

How can I factorize this quadratic expression

Going by the exercises of a book I have been factorizing quadratic equations the following way, let's say I have: $$ {x^2 - 7x + 12 = 0} $$ I know that $$ {a \times b = 12 \\ \text{ and } \\ a + b ...
6
votes
2answers
324 views

quadratic equation what am I doing wrong?

solve $$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} $$ $$ \sqrt{5x+19} = \sqrt{x+7} + 2\sqrt{x-5} \Rightarrow $$ $$ 5x+19 = (x+7) + 4\sqrt{x-5}\sqrt{x+7} + (x+5) \Rightarrow $$ $$ 3x + 17 = ...
-2
votes
1answer
47 views

Solving a quadratic involving square root

$$\sqrt{\frac{x}2} = 1-x$$ so $$x = ?$$ I have tried to solve many times and i got $x = \frac52$ everytime. But my book says answer is $\frac12$. I think i couldn't understand square roots clearly.. ...
-3
votes
2answers
50 views

Show that $b^2=9a^2+4ac$ given that the square roots differ by 3 for a quadratic equation $ax^2+bx-c=0$ [closed]

If the square roots of $ax^2+bx-c=0$ differ by 3, show $b^2=9a^2+4ac$. How do I show this, from the knowledge that I have about alpha and beta, I can't get it right.
0
votes
4answers
36 views

Finding value of $m$ which is a part of a quadratic

Q : $$m \gt 2$$ $$x^2 + (m-3)x - 2 = 0$$ If $|x_1 - x_2| = 3$, so $m = ?$ Stuck here. Please give me a hint
2
votes
3answers
241 views

Solving an operation involving roots of a quadratic equation

A question from my book: $3x^2 + 7x + 5 = 0$ So, $\sqrt{(x_1^2 + 2x_1x_2 + x_2^2)} + x_1x_2 = ?$ Options: A) $4$ B) $5$ C) $6$ D) $7$ E) $8$ It's looking too easy, my answer is $-\frac{2}3$, but it ...
0
votes
2answers
50 views

An equation with negative exponents in quadratic equations test

There is a problem like this : $x^{-1} = 2x^{(-1/2)} + 3 , x = $? in my test. I'm working on it for a half of hour but still i can't solve. Please help me. (Excuse my bad grammar)
0
votes
1answer
21 views

Finding equation of parabola when given point and tangent line

I have tried substituting the coordinate points (2,4) and the formula for the turning point (-b/2a, etc.) into the equation and solving it simultaneously, however this has not seemed to work. The ...
1
vote
3answers
52 views

$ax^2+bx+c=0$ has roots $x_1,x_2$. what are the roots of $cx^2+bx+a=0$.

Given solution: Dividing the first equation by $x^2$ we get $c(\frac{1}{x^2})+b(\frac{1}{x})+a=0$ so $(\frac{1}{x_1}),(\frac{1}{x_2})$ are the roots of $cx^2+bx+a=0$.{How?It is not obvious to me.} ...
2
votes
2answers
52 views

Help me to prove this statement about quadratic equations? (from Gelfand's Algebra).

$ x^2+px+q=0 ${p,q are integers; a,b are roots}. Prove $a^n+b^n$(n is any natural number) is an integer. This is the third part of the problem.I have previously proved that $a^2+b^2$ and $a^3+b^3$ ...
0
votes
2answers
62 views

Difference between two real roots with uniformly distributed coefficents

I have a question that first I need to know what is happening, but then I also need to code it in a program called APPL, which is an extension from Maple18 that I really have never used, yet I have ...
0
votes
2answers
60 views

calculate the intersection of two number series

I have a series of numbers. It is in the form of a parabola. This series is guaranteed to have at least one perfect square within it (edited I thought there was only one). The second series is also a ...
1
vote
3answers
29 views

Relationship between roots and equations

I'm stuck on topic of relationship between roots and equations. The roots of $x^2 -2x +3 =0$, are $\alpha$ and $\beta$. Find the equation whose roots are : 1- $\alpha+2$, $\beta+2$ 2- $\alpha^2$, ...
7
votes
4answers
296 views

Why are there four solutions to $x^2-2x-8=0$ in $\mathbb{R}$? Or am I wrong?

It might be a very trivial question to ask but why do we get four different solutions for a quadratic equation using these two methods? $x^2-2x-8=0$ We see that factors are $(x-4)$ and $(x+2)$ so ...
2
votes
2answers
48 views

How do I complete the square when the $x^2$ has a coefficient greater than $1$?

For homework we are doing completing the square and a few of them have coefficients greater than one. For example one of the quadratic equations we have to complete the square of is $-2x^2-7x-2$. All ...
0
votes
1answer
25 views

Complex Roots Of a equation - Equilateral triangle

$z_1$ and $z_2$ are the roots of $3z^2+3z+b=0$.If $O(0),A(z_1),B(z_2)$ is an equilateral triangle then what will be the value of b ? My approach:I took $z_1=m_1+in_1$ and $z_2=m_2+in_2$ and proceeded ...