All Questions
Tagged with cubics quadratics
55
questions
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Solving a mixed system of 2 cubic and 2 quadratic equations with 4 unknowns
I tried plugging these cubic and quadratic equations into Wolfram Alpha and Symbolab but both said the same thing, too much computing time required. Now I am struggling to solve these equations and I ...
- 1
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cubic equation edge cases
Working on general cubic equation solver in form ax^3+bx^2+cx+d=0 And have no clue for special cases:
In terms of cubic there should be one real root and two complex, or 3 real roots if coefficients ...
2
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3
answers
94
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Will the perpendicular bisector between the line connecting two cubic roots of the same arc never intersect its turning point?
The quadratic graph: $$ f(x) = (x+2)(x+1)$$
would have a midpoint between its roots at $x = -1.5$. This line would intersect its turning point.
However the cubic graph: $$ f(x) = (x+1)(x-2)(x+3)$$
...
- 51
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If $ax^3+bx-c$ is divisible by $x^2+bx+c$ then $a,b,c$ are in what kind of progression? (arithmetic/geometric/etc) [closed]
If a polynomial $$f(x)=ax^3+bx-c$$ is divisible by the polynomial $g(x)=x^2+bx+c$, then $a,b,c$ are in ...
$1.$ Arithmetic Progression
$2.$ Geometric Progression
$3.$ Harmonic Progression
$4.$ ...
- 1,253
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When we say the odd degree polynomial has odd number of real roots, is there any condition on the coefficients?
I read that if the degree of a polynomial equation is odd then the number of real roots will also be odd.
I took the example of a cubic equation. If it has imaginary roots then that will occur in pair....
- 8,338
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3
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Solve the equation $x^3-11x^2+38x-40=0$, given that the ratio of two of its roots is $2:1$.
Solve the equation $x^3-11x^2+38x-40=0$, given that the ratio of two of its roots is $2:1$.
By hit and try, I can see that $x=2$ is a root.
Dividing the given cubic by $x-2$, I get a quadratic whose ...
- 8,338
3
votes
3
answers
214
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Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
My Attempt:
On rearranging, I get, $(x^2+x+1)(3x+7)+2=0$
Or, $3x^3+10x^2+10x+9=0$
Derivative of the cubic ...
- 8,338
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Area (quadratics) go negative vs. Volume (cubics) goes to infinity
Consider a typical "fenced garden" problem, which is easily represented by a quadratic, e.g.,
$a(w)=-10w^{2}+100w$ where $w$ is the width (or the wall, if you prefer).
which peaks at $(5,250)...
- 113
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A cute way to solve the quadratic. How to extend it to the cubic?
Playing around with complex numbers, I found a cute way of solving the quadratic equation.
Let's start with the (monic) equation
\begin{equation}
z^2+pz+q = 0
\end{equation}
where $z$ and the ...
- 3,386
1
vote
3
answers
84
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Assume that $a=b$. If the function $f(x)$ is monotonously increasing, then (answer 1) $0<a<1$ (answer b).
The given function is $f(x)=x^3-3ax^2+3bx-2$.
I am aware that monotonously increasing means to continuously increase, so I tried getting this function's derivative and then setting it to zero, but to ...
- 149
15
votes
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answer
354
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Does the limit of the cubic formula approach the quadratic one as the cubic coefficient goes to $0$?
The formula for solving a cubic equation of the form $ax^3+bx^2+cx+d=0$ does not seem to yield the quadratic formula for the limit $\lim _{a \rightarrow 0} \text{(cubic formula)}$.
But, if one tries ...
- 297
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What properties do have odd quadratic parabola and even cubic parabola? Were they ever described?
Here is a normal quadratic parabola:
$f_{even}=\frac{x^2}{4}+\frac{\pi^2}6$
Here is an odd parabola:
$f_{odd}= \text{Li}_2\left(e^x\right)+ x \log \left(1-e^x\right)-\left(\frac{x^2}{4}+\frac{\pi ^2}{...
- 9,261
2
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2
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207
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Sum of the first $n$ quadratic integers
I'm from Italy so maybe my English could be bad. To prove that
$$
S=\sum_{k=1}^{n} k^2 = \frac{n(2n+1)(n+1)}{6}
$$
we can consider the series:
$$
\sum_{k=1}^{n}\left((k-1)^3-k^3\right) \, .
$$
Note ...
- 21
-1
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2
answers
198
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Cubic's derivative
For a cubic $ax^3+bx^2+cx+d$, the radicand of the derivative is $b^2-3ac$ and I was wondering if there's a word for this term. I know that the nature of it being positive, zero, or negative will ...
user612996
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A polynomial has the same remainder when divided by $x+k$ or $x-k$; what is $k$?
Question
Given that $y = 3x^3 + 7x^2 - 48x + 49$ and that $y$ has the same remainder when it is divided by $x + k$ or $x - k$, find the possible values of $k$.
My attempt
Let $f(x) = 3x^3 + 7x^2 - ...
- 49