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 $y=a(x+b)(x+c)$.

learn more… | top users | synonyms (1)

1
vote
2answers
523 views

If $ax^2+bx+c=0$ and $2x^2 +3x+4=0$ have a common root where $a,b,c \in \Bbb N$,find least value of $a+b+c$

Problem: If $ax^2+bx+c=0$ and $2x^2 +3x+4=0$ have a common root where $a,b,c \in \Bbb N$, find least value of $a+b+c$ Solution: Here $2x^2 +3x+4=0$ will give complex roots These roots will ...
2
votes
3answers
120 views

Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
1
vote
0answers
24 views

Geometric interpretation of the coefficients of the quadratic equation.

The quadratic equation has three general forms: $ax^2+bx+c$ $a(x-r_1)(x-r_2)$ $a(x-h)^2+k$ $r_1$ and $r_2$ are the zeroes of the quadratic. $h$ is the horizontal position of the ...
0
votes
4answers
67 views

How do I find the solution(s) to my second-degree equation?

$$f(x) = x^2 - 3x$$ My attempt : $$ \begin{align} x^2-3x &= 4\\ x(x-3) &= 4\\ x-3 &= 4 \\ x &= 7\\ \end{align} $$ I managed to solve one part of this problem but that one part is ...
2
votes
5answers
84 views

When is $\sqrt{x/y^2}$ equal to $\sqrt{x}/y$?

The solution to the quadratics is given by $r = -\dfrac{b}{2a}\pm\sqrt{\dfrac{b^2-4ac}{4a^2}}$, which is shortened to $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$, but I'm wondering how if this ...
-10
votes
0answers
34 views

Quadratic term answer fast please [on hold]

let k be a real number such that k≠0.if a and b are non zero complex numbers satisfying a+b=-2k and a²+b²=4k²-2k,then a quadratic equation having (a+b/a)and (a+b/b)as its roots is equal to
-8
votes
0answers
37 views

Quadratic term answer fast [on hold]

let sin x and sign y be the roots of the quadratic equation a sin square theta + b sin theta + C is equal to zero,such that sin x + 2 Sin y is equal to 1 then the value of a square + 2bsquare + 3ab+ ...
0
votes
2answers
45 views

How to solve the following quadratic word problem given a quadratic equation?

The height of a ball(h), in feet, after s seconds is modeled by the equation $$h=-16t^2+40t-6$$ How many seconds does it take for the ball t reach its maximum height? First thing i did was turn the ...
3
votes
2answers
40 views

When are we able to find a quadratic with roots that are a function of another quadratic?

Motivation: Given the roots of the quadratic $2x^2+6x+7=0$ find a quadratic with roots $\alpha^2-1$ and $\beta^2-1$ I was able to solve this problem in two ways: Method 1: Sum of the roots $\alpha+...
2
votes
3answers
58 views

Concept of roots in Quadratic Equation

$a$ , $b$, $c$ are real numbers where a is not equal to zero and the quadratic equation \begin{align} ax^2 + bx +c =0 \end{align} has no real roots then prove that $c(a+ b+ c)>0$ and $a(a+ ...
-1
votes
1answer
19 views

Find the range of values for k for which the equation $x^2+(a-2)x+(a+3)=0$ had no real roots

I don't understand the question at all. Find the range of values for a for which the equation $x^2+(a-2)x+(a+3)=0$ has no real roots.
0
votes
3answers
46 views

Why are the solutions of the equation different? : $x=2 => x^2=4 => x=±2$

If I define the variable $x$ as $x=2$, then $x^2=4$. But the solutions of $x^2=4$ are $±2$(two solutions). I defined what the variable $x$ is, then why are the solutions for the equation $x^2=4$ two, ...
1
vote
1answer
86 views

What technique reduces factorable ax^2+bx+c=0 to factorable where a=1

The normal way to factor ax^2+bx+c=0 is to look for t,u,v,w such that: (tx+u)(vx+w) = 0 so that tv=a, uw=c, and uv+wt=b. This can be tricky, since there can be several possibilities for t,u,v,w. ...
-5
votes
5answers
59 views

On the equation $|x|^2+|x|-6=0$

Which of the following are true for $$|x|^2+|x|-6=0$$ 1. It has $4$ roots 2. The sum of the roots is $-1$ 3. The product of the roots is $-4$ 4. The product of the roots is $-6$ Only one of the ...
1
vote
3answers
79 views

How to get the correct angle of the ellipse after approximation

I need to get the correct angle of rotation of the ellipses. These ellipses are examples. I have a canonical coefficients of the equation of the five points. $$Ax ^ 2 + Bxy + Cy ^ 2 + Dx + Ey + F = 0$...
2
votes
4answers
184 views

Is it really necessary to learn how to write a quadratic in standard to vertex form?

How crucial is this skill or form of writing and polynomial function? Can't we just always use the $-b/2a$ trick for the $x$ intercept and just plug it back into the function to find the $y$?
3
votes
4answers
102 views

Solve $2^x+4^x=2$

This is the equation, but the result is different from wolframalpha: $$2^x+4^x=2$$ $$2^x+2^{2x}=2^1$$ $$x+2x=1$$ $$x=\frac{1}{3}$$ WolframAlpha: $x=0$ Where is the error?
0
votes
1answer
34 views

If the volume of a container is $196~\text{cm}^3$, find the dimensions of the original template.

This is a quadratics problem. The full question reads: An open container with a square base is made by cutting $4~\text{cm}$ square pieces out of a piece of tin. If the volume of the container is $...
-2
votes
2answers
43 views
1
vote
2answers
61 views

How to solve for log with a number outside?

$$\log_6(4x-10)+1 = \log_6(15x+15)$$ This is a sample problem. I know that when the bases of log are the same, all you have to do is set the parenthesis inside equal to each other. If the $1$ wasn't ...
0
votes
2answers
30 views

Find all values of parameter a, when sum of solutions of following equation is 100

Find all values of parameter $a$, when sum of solutions of following equation is $100$. $$ \sin(\sqrt{ax-x^2})=0 $$ I tried to get rid of that $sin$ and there was quadratic equation with two ...
2
votes
5answers
57 views

Solving $2x^4+x^3-11x^2+x+2 = 0$ [duplicate]

I am having no idea how I can solve this problem. I need help! Here's the problem $2x^4+x^3-11x^2+x+2 = 0$ I am learning Quadratic Expressions and this is what I need to solve, and I can't ...
-1
votes
1answer
14 views

Help with some calculations

My question is: what I need to do to get 2nd equation from the first? 1) $TP1 = vp1 · λ + TS1$ $TP2 = vp2 · λ + TS2$ 2)$$TP_2 − TS_2 =\frac{vp2}{vp1}(TP1 − TS1)$$
0
votes
3answers
62 views

Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k$

A quadratics question. Show algebraically that the graph of $y=x^2 + kx - 2$ will cut the $x$-axis twice for all values of $k.$ I recently asked a similar question, but this problem seems ...
3
votes
5answers
89 views

The roots of the equation $x^2 - 6x + 7 = 0$ are $α$ and $β$. Find the equation with roots $α + 1/β$ and $β + 1/α$.

Quadratic equation question, as specified in the title. The roots of the equation $x^2 - 6x + 7 = 0$ are $α$ and $β$. Find the equation with roots $α + \frac{1}{β}$ and $β + \frac{1}{α}$. I ...
1
vote
1answer
53 views

For what values does $kx^2 - 6x - 4 = 0$ cut the x-axis? [closed]

A quadratic equation problem. For what values does $kx^2 - 6x - 4 = 0$ cut the x-axis? The preceding part of this question dealt with the discriminant.
1
vote
1answer
41 views

Limit of the root of quadratic equation

The root of the equation $ a x^2 + bx + c = 0 $ is given by $$ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \;\;\;...(1) $$ On the other hand, if $a = 0$, then from the original equation we get $$ x = - \...
0
votes
1answer
870 views

How do I prove a quadratic is always positive or negative for x?

I looked this up and seen something that was beyond my A-Level Maths course. In class we are doing the discriminant and sketching quadratic graphs, so it is nothing advanced. My teacher completed ...
1
vote
6answers
61 views

How to solve without solving by inspection? $\frac{x+5}{x+k}=\frac{-kx+5}{x-1}$

Background: This is from a test review on functions. The original problem was Find the value of $k$ so that the function $f(x) = \frac{x+5}{x+k}$ will be its own inverse. I found the answer by ...
1
vote
2answers
57 views

Solve the quadratic equation

$$ \sqrt{a-\sqrt{a+x}}=x $$ This equation contains one variable x we have to find the value of x.I tried to simplify it but it doesn't work....i have also tried the basic concepts of quadratics ...
3
votes
2answers
799 views

Optimization: maximum area of a triangle under a parabola

Optimization: maximum area of a triangle in a parabola Inside a curve ($x^2-25$ - Parabola) a triangle is drawn with A as the vertex at the origin and the line joining points B and C lie on the ...
1
vote
2answers
44 views

Quadratic inequality proof.

I recently encountered a quadratic equations property that $ax^2+bx+c>0$ $ \forall$ $x\in \Re \Rightarrow D<0$ $and$ $a>0$ and $ax^2+bx+c<0$ $ \forall$ $x\in \Re \Rightarrow D&...
1
vote
1answer
34 views

System of two Nonlinear equations

I have a probably very simple problem here. A system of nonlinear equations. $$\left\{ \begin{align} & {{x}^{2}}+{{y}^{2}}=26 \\ & x+{{y}^{2}}=6 \\ \end{align} \right.$$ I started with ...
0
votes
1answer
70 views

Permuting the roots of a cubic polynomial with a quadratic polynomial cyclicaly

The polynomial $Q(x)=x^3-21x+35$ has three distinct real roots $r,s,t$. Find reals $a,b$ so that $P(x)=x^2+ax+b$ satisfies $P(r)=s,P(s)=t,P(t)=r$ or $P(r)=t,P(t)=s,P(s)=r$. I tried using cardano to ...
0
votes
2answers
46 views

how to solve a trignometric quadratic equation?

I am stuck in a question... the questions says $sin^4 x -(k+2)sin^2 x -(k+3)=0$ has a solution then what is the interval in which k must lie. I tried to solve it by putting $sin^2 x =p$ then putting ...
1
vote
2answers
82 views

Solution of given equation for $x$

Solve the given equation for $x$ $$\sqrt{x^2-2x+8}+\sqrt{x^2-2x+3}=125$$ I solved the question by taking ${x^2-2x+3}=t$, and squaring twice and finally solving ${x^2-2x+3}=t$ but it required very ...
0
votes
1answer
54 views

What is the equation of the quadratic function through $(2,5)$ with roots $1+\sqrt 5$ and $1-\sqrt 5$? [closed]

Determine the equation of the quadratic function that passes through $(2,5)$ if the roots of the corresponding quadratic equation are $1+\sqrt 5$ and $1-\sqrt 5$.
0
votes
5answers
77 views

Solve the following $\frac{3x}{x+6} \ge 0 $

Solve $$\frac{3x}{x+6} \geq 0 $$ My work $$(x+6) / 3x <0 $$ $$1/3 + 6/x <0 $$ $$ 6/x <-1/3 $$ $$ x >-18 $$ is that correct
2
votes
3answers
58 views

Solving $x^2 - 16 x+55> 0$ for $x$

Solving $x^2 - 16 x+55> 0$ for $x$ my work $$(x-11)(x-5) > 0$$ then x >11 and x > 5 is that correct ???
1
vote
1answer
36 views

quadratic function vs conic section

I am categorizing types of math problems on the ACT. I started off with 'quadratic function' as one category, and 'conic sections' as another... It seemed like a simple classification at first, but ...
4
votes
2answers
79 views

For $ax^2+bx+c$ prove that $|a|+|b|+|c|\leq 17$

Let $ax^2+bx+c$ be a quadratic polynomial with real coefficients such that $$|ax^2+bx+c| \leq 1,$$ for $ 0\leq x\leq 1$. Prove that $$|a|+|b|+|c|\leq 17$$ How to proceed in this particular question. ...
4
votes
10answers
156 views

Factor $6x^2​ −7x−5=0$

I'm trying to factor $$6x^2​ −7x−5=0$$ but I have no clue about how to do it. I would be able to factor this: $$x^2-14x+40=0$$ $$a+b=-14$$ $$ab=40$$ But $6x^2​ −7x−5=0$ looks like it's not ...
0
votes
0answers
28 views

Finding integer solutions to quadratics in the form [duplicate]

In a set containing two different types of elements the probability of randomly choosing two elements of the same type can be expressed as: $$\ \frac nm * \frac {n-1}{m-1} = \frac 1x$$ Where n is ...
1
vote
1answer
55 views

Are there more quadratics with real roots or more with complex roots? Or the same?

Consider all quadratic equations with real coefficients: $$y=ax^2+bx+c \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,, a,b,c \in \mathbb{R}, \, a≠0 $$ I was wondering whether if more of them have real roots, more ...
0
votes
1answer
20 views

Vertex (smallest possible value) of $ax^2+bx+c$

The original problem was this: Find the smallest possible value of $ax^2+bx+c$, where a, b, and c are given numbers and $a>0$, and x is some number. I already asked this, and got a decent answer, ...
1
vote
1answer
15 views

The solution set of a multivariate quadratic after transformation of variables

I have an equation given in terms of three vectors $\vec{x}$, $\vec{y}$ and $\vec{z}$, all in $\mathbb{R}^n$: $$1 - (\vec{x}\cdot\vec{y})^2 = K + 2K\vec{z}\cdot\vec{y} + (\vec{z}\cdot\vec{y})^2, K \...
0
votes
1answer
479 views

Find pressure in a sinusoidal function

Tiffany is a model rocket enthusiast. She has been working on a pressurized rocket filled with laughing gas. According to her design, if the atmospheric pressure exerted on the rocket is less than 10 ...
0
votes
1answer
36 views

How can I find the largest possible subset A of $\mathbb{R}$?

So I have this equation $$f(x)= \frac{x^2}{(x-2)(x+3)}$$ and I need to find the largest possible subset $A$ of $\mathbb{R}$ that could form the domain of a function. Can anybody help me? I really don'...
1
vote
2answers
41 views

Determine if quadratic diophantine equation in two variables will generate perfect squares

I have come across two equations with variables $x,y$ \begin{align*} (x+ay)^2+ 4 x y\\ (x-y)^2-4 c x y \end{align*} where $a,c\in \mathbb{Z}_+$ are some constants. I would like to determine the ...
3
votes
3answers
105 views

Need help solving $x^4-3x^3-11x^2+3x+10=0$

Solve $x^4-3x^3-11x^2+3x+10=0$ I have tried to solve this equation using 'general formula from roots' from https://en.wikipedia.org/wiki/Quartic_function. $$ax^4+bx^3+cx^2+dx+e=0$$ $$x_{1,2}=-\frac ...