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

-1
votes
1answer
21 views

Quadratic Equation roots when there is a constant [on hold]

What are the roots of x2+x-2n where n is a constant? Can someone help me with a step by step approach?
0
votes
1answer
13 views

Guarantees of a Quadratic Form for Two Related Matrices

Suppose I have a matrix: $\mathbf{Q} = \mathbf{Z}^T\mathbf{B}\mathbf{Z}$ where $\mathbf{Z} \in \mathbb{R}^{n\times m},\mathbf{B}\in \mathbb{R}^{n\times n}, m \leq n$ Can I say that if $\mathbf{Q}$ ...
0
votes
0answers
33 views

Solve the following equation to find value of $64xyz$ [on hold]

If the real numbers $x,y,z$ are such that $x^2 + 4y^2 + 16z^2 = 6xy + 4yz + 2zx =3$. Then find value of $64xyz$
1
vote
3answers
40 views

If $ V_n= \alpha^n+\beta^n$ and $\alpha,\beta$ are roots of $x^2+x-1=0$, then $V_n+{V}_{n-3}=2{V}_{n-2}$?

If $ V_n= {\alpha}^n+{\beta}^n$, where ${\alpha}$ and ${\beta}$ are roots of the equation $x^2+x-1=0$. Then prove that $V_n+{V}_{n-3}=2{V}_{n-2}$ (n is whole number). I have tried to manipulate ...
2
votes
0answers
53 views

quadratic formula returning one real solution when none should exist

i rely a lot on the quadratic formula, but i'm learning to grow a little wary of it. or of how i use it. for example if the "A" term is zero (ie your curve is a line) then the quadratic divides by ...
1
vote
4answers
52 views

Rational Roots of a Quadratic Equation

If $a,b,c$ are non zero, unequal rational numbers then prove that the roots of the equation $$(abc^2)x^2+ 3a^2cx+b^2cx-6a^2-ab+2b^2=0 $$ are rational. Use theory of equations and basic ...
0
votes
0answers
60 views

Factoring quartic into 2 quadratic polynomials: $x^{4}+ax^{3}+bx^{2}+cx+d =(x^{2}+g_{1}x+h_{1})(x^{2}+g_{2}x+h_{2})$

I would like to factor the quartic into two quadratic polynomials $F(g),G(h)$: \begin{align*} x^{4}+ax^{3}+bx^{2}+cx+d & =(x^{2}+g_{1}x+h_{1})(x^{2}+g_{2}x+h_{2}),\\ & =x^{4}+(g_{1}+g_{2})x^{...
-2
votes
0answers
44 views

How to know WHEN you should Factor? [on hold]

so I have an issue when it comes to factoring...How do you instinctively, know when factoring comes into play? Ive come across a few instances and times when I was working on a problem, and I was ...
0
votes
0answers
31 views

How to reverse an equation [on hold]

I have an equation describing how incubation temperature ($t$) relates to the incubation period ($IP$) of sea turtle eggs (as eggs in warmer sand incubate faster): $$IP = 59.0785 - 5.3805t + 0.5482t^...
0
votes
1answer
22 views

Parameterization of Equation with Gaussian Integer

I'd like to ask how to get the parameterization to this equation: $3z_1^2+z_2^2=156$, where $z_1$ and $z_2$ are both Gaussian integers. More generally, is there any parameterization to the general ...
2
votes
1answer
36 views

Imaginary Roots of quadratics and Graph

I have this equation: $x^2-4x+5 = 0$ . Its roots are imaginary $2 \pm i$ and I read in Algebra book/resource somewhere that to graph a quadratic equation with imaginary/complex roots, you need a ...
0
votes
1answer
25 views

Prove that the quadratic equation is bijective

Given $$f(x)=ax^2+bx+c\ ; \quad a\neq0.$$ Prove that it is bijective if $$x \in \Bigg[\frac{-b}{2a},\ \infty \Bigg]$$ and $$ranf=\Bigg[\frac{4ac-b^2}{4a},\ \infty \Bigg).$$ I can prove that the ...
1
vote
1answer
58 views

What does the floating point number $(1.2)$ down the bracket mean?

What does the 1.2 on the image mean? Thanks in advance.
-4
votes
0answers
22 views

Parameters of a Quadratic Function [closed]

Calculate $a$ and $b$ so that the function graph $y= ax^2 + bx+6$ has the vertex at the point $(5/2, -1/4)$.
0
votes
4answers
68 views

What is the equation of x= $\sqrt2$ and $-\sqrt2$ for $ax^2+bx+c=0$? [closed]

I try to find a quadratic equation having solutions $\sqrt2$ and $-\sqrt2$ . I know it is $x^2-2$ but it must be also "b" value for x so I could not find this yet.
1
vote
1answer
13 views

Number of non negative integral values of $n,n\le 10$ so that a root of the equation $n^2\sin^2 x-2\sin x-(2n+1)=0$ lies in the interval $[0,\pi/2]$

Find the number of non negative integral values of $n,n\le 10$ so that a root of the equation $n^2\sin^2 x-2\sin x-(2n+1)=0$ lies in the interval $[0,\pi/2]$ As $x\in[0,\pi/2]$ so $\sin x\in[0,1]$ ...
0
votes
2answers
26 views

Find the least value of $n\in N$ for which $(n-2)x^2+8x+n+4>\arcsin(\sin12)+\arccos(\cos12)$ for every $x \in \mathbb {R}$

Find the least value of $n\in N$ for which $(n-2)x^2+8x+n+4>\arcsin({\sin12})+\arccos({\cos12})$ for every $x \in \mathbb {R}$ $(n-2)x^2+8x+n+4>\arcsin({\sin12})+\arccos({\cos12}) $ $(n-2)x^2+...
1
vote
3answers
82 views

Quadratic equation sum from Russian book [duplicate]

Solve $\sqrt{5-x}=5-x^2$ for $x$. This is what I have done so far. Method 1: \begin{align} \sqrt{5 - x} & = 5 - x^2 \\ 5 - x & = (5 - x^2)^2 \\ 5 - x & = 25 - 10x^2 + x^4 \\ 0 & = x^...
0
votes
1answer
54 views

Does there exist a formula to find the coefficients of this parabola?

For equation $y=ax^2+bx+c$, assume I know the value of the coefficient $a$, I know the value of $y$ at the parabola's vertex (though I do not yet know the $x$ at that point), and I am given a point $(...
0
votes
1answer
43 views

If the roots of the equation $ax^2-2bx+c=0$ are complex ,then find the number of real roots of the equation $4e^x+(a+c)^2(x^3+x)=4b^2x$.

If the roots of the equation $ax^2-2bx+c=0$ are imaginary,then find the number of real roots of the equation $4e^x+(a+c)^2(x^3+x)=4b^2x$. The only information i'm able to interpret is$-$ Since the ...
1
vote
1answer
39 views

Quadratic equation_to prove that the roots of equation are rational

If $a, b, c$ are rational, show that the roots of the equation $abc^2x^2 + 3a^2cx + b^2cx – 6a^2 – ab + 2b^2 = 0$, are rational.
0
votes
4answers
50 views

Let $P(x)=x^2+bx+c$, where $b$ and $c$ are integers.

Let $P(x)=x^2+bx+c$, where $b$ and $c$ are integers. If $P(x)$ is a factor of both $f(x)=x^4+6x^2+25$ and $g(x)=3x^4+4x^2+28x+5$, then $P(x)=0$ has imaginary roots $P(x)=0$ has roots of ...
0
votes
9answers
58 views

Common root question on quadratics equations to show that $a+b+c=0$

If $f(x)=ax^2+bx-c$ and $g(x)=ax^2+cx+b$ have a common root , show that $a+b+c=0$. I tried this by thinking that $\alpha$ is the common root and then I got by substituting and solving , $(b^2+c^2)(b-...
0
votes
1answer
37 views

Quadratic forms, change of variables

If one has a symmetric matrix $A$, one can diagonalize it with an orthonormal change of basis vectors, e.g. $S^TAS$ is diagonal. Now lets consider the following matrix $$A=\begin{bmatrix} 1&1\\ 1&...
1
vote
4answers
62 views

Find the least value of the expression $x^2+2xy+2y^2+4y+7$

Find the least value of the expression $x^2+2xy+2y^2+4y+7$ I am not able to solve this equation though i am able to differentiate it.
3
votes
8answers
120 views

Show that $x^2+9x+20$ is divisible by 2 for all $x \in \mathbb{Z}$

I'm having extremely hard time getting how proof by induction should work for this case. This is my attempt so far: (1) When $x = 1$ $1^2 + 9 + 20 = 30$ which is divisible by 2. (2) Now assume ...
1
vote
2answers
76 views

How to plot complex poles/zeros into Bode plot?

I am wondering how to plot complex poles/zeros into Bode plot (signal magnitude vs. circular frequency (or $j\omega$)). Complex poles/zeros differ from simple poles/zeros in such way that complex ones ...
1
vote
2answers
45 views

If $c,d$ are the real roots of the equation, $(x-a)(x-b)=f$

EDIT: Sorry, corrected the Typo If $c,d$ are the real roots of the equation, $$(x-a)(x-b)=f$$ Then roots of the equation, $$(x-c)(x-d)+f=0$$ are? $a,b$ $\frac{a}{f},\frac{b}{f}$ $\frac{f}{a},\...
0
votes
4answers
34 views

Quadratic equation with roots

The quadratic equation $2x^2 + 7x + 5 = 0$ has a roots $α$ and $β$. Form a quadratic equation with roots $3α$ and $3β$.
1
vote
1answer
25 views

Determining the equation of $g(x)$ given intersecting points via another function?

Here's the problem: I know this is a system of quadratic-quadratic equations. My attempt was to convert the equation of $f(x)$ into the form $f(x) = a(x - p)^2 + q$, making the $a$ negative to ...
1
vote
1answer
28 views

Solving quadratic inequality to get the final answer

Okay, so I'm self studying my A level mathematics and I don't have a tutor to ask from so I hope someone can help me here. My question is: Q: The equation of curve is $y=\dfrac{12}x$ and the of a ...
0
votes
1answer
31 views

How to transform Parabola to 3D graph and lift the vertex upon a vertical line (z axis)?

If I have a quadratic in the form of $f(x) =ax^2+bx+c$, how would I be able to transfer this onto a 3D Graph where $x$=horizontal, $y$=the other 2D dimension and $z$= height , and raise the turning ...
1
vote
1answer
29 views

For which values of $k$ can $( x +y +z)^2 + k(x^2 +y^2 +z^2)$ be resolved into linear rational factors?

My first attempt : Tried to solve by polynomial formulas but can't proceed after few steps. Second attempt : Tried by vectors but found nothing useful.
0
votes
2answers
47 views

Common point between ellipse and tangent passing through external point

Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and a point $(u, v)$ not on the ellipse, I want to find two points that lie on the ellipse and on the two tangents of the ellipse passing ...
-1
votes
0answers
29 views

how to solve arithmetic sequence (find the N) with quadratic equation

how to solve arithmetic sequence (find the N) with quadratic equation A= 75 (price per floor) D=3.70(price increase per floor) N=? (what floor that cost $$2820.50 within a couple of Dollar) and ...
2
votes
1answer
203 views

Really universal quartic formula

At the outset, I would like to say hello to the honorable discussants in this forum. [This is my first entry in this forum, so I apologize in advance for any possible mess.] I have a need to write ...
0
votes
3answers
38 views

theory of equation..finding roots of a modular function

The number of real roots of $\big|x^2+4|x|+3\big| +2x-11=0$ is? I have tried by expanding modulus, and if I'm right I'm getting two real roots but I'm a bit confused because of the innermost modulus, ...
0
votes
2answers
40 views

Vectors, calculate distance from these two points [closed]

My Attempt Thus far.. I tried using the quadratic formula on this to find the distance/magnitude , however it did not work. I then tried to solve it as a quadratic inequality. Can anyone help? And ...
2
votes
1answer
41 views

Transforming a quadratic from its zeroes

In below problem I've been trying to figure out how simply letting $y=\dfrac{x}{x+10}$ gives the quadratic with roots scaled by $\dfrac{1}{\alpha+10}$. I'm a bit clueless why it works. My thoughts : -...
2
votes
4answers
174 views

Relevance of Complex roots of Quadratic Equation

Let's say Amy is a stunt pilot, planning on doing a parabolic dive in an air show: $y = x^2 + 4x +5$ She hopes to use this trajectory to dive close to the ground (the x-axis, height is the y-axis), ...
4
votes
3answers
278 views

How many solutions are there for $a^b = 1$

I want to find out the solutions to the equation $a^b = 1$. I know the real solutions like $k^0 = 1, 1^k = 1, (-1)^{2k} = 1$. I want to know if I missed any real solutions for this. Also I know of ...
0
votes
1answer
33 views

How to solve quadratic equation problem having a prepositional logic?

When $\alpha,\beta$ are roots of $x^2+bx+c=0$, Find the equation whose roots are $p$ and $q$ where, $p=\alpha +\beta^2,\,q=\beta+\alpha^2$. Also when $\alpha,\beta$ are imaginary show that, $b=-1\,$...
0
votes
3answers
49 views

Solving a quadratic equation without expanding

Solve $(t-1)^2 + 4(t-1) -12 = 0$ without expanding. My first thought was to solve by expanding $(t-1)^2$ to $(t+1)(t-1)$ and then divide by $(t-1)$, bu obviously that leaves $\frac{-12}{t-1} $. I ...
2
votes
5answers
75 views

How is discriminant related to real $x$?

Question in my text book Solve for range of the function, $$y=\frac{x^2+4x-1}{3x^2+12x +20}$$ Text book says, cross multiply and express the obtained equation as a quadratic equation in $x$ So I ...
4
votes
3answers
57 views

Trigonometry and quadratics : Possible mismatch?

There’s this problem I came across, gives me an invalid answer by using general quadratic formula. Wonder why? $2\sin^2{x} -5\cos{x} -4 =0 $ Here’s what I did: $2\sin^2{x} -5\cos{x} -4 =0 $ $2(1-\...
10
votes
2answers
116 views

Prove that polynomial of degree $4$ with real roots cannot have $\pm 1$ as coefficients (IITJEE)

So I was going through my 11th class package on Quadratic equations and I saw a question to prove that a polynomial of $4$th degree with all real roots cannot have $\pm 1$ as all its coefficients. ...
5
votes
7answers
204 views

Minimum value of $\frac{b+1}{a+b-2}$

If $a^2 + b^2= 1 $ and $u$ is the minimum value of the $\dfrac{b+1}{a+b-2}$, then find the value of $u^2$. Attempt: Then I tried this way: Let $a= bk$ for some real $k$. Then I got $f(b)$ in ...
0
votes
2answers
46 views

Applying quadratic equations.

So I realise this is quite an easy question, but for some reason I can't see the solution. So the question followed on from a previous question where we used a quadratic equation to find the ...
5
votes
2answers
112 views

Reference request: Transformations under which the discriminant is invariant

I had an occasion to think about this quadratic equation: $$ ax^2 +bx(1-x) + c(1-x)^2 = 0. $$ Its solution is $$ x = \frac{2c-b\pm\sqrt{b^2-4ac\,}}{2(a-b+c)}. $$ The thing under the radical is the ...
2
votes
4answers
93 views

Solving this quadratic equation by completing the square?

$-\frac{2}{3}x^{2} - x +2 = 0$ Here's what I did: However, the textbook answer is $x = -2.6, x = 1.1$. What did I do wrong?