# Questions tagged [completing-the-square]

Questions on the algebraic operation of completing the square. Should probably be used with the (algebra-precalculus) tag.

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### Prove that for $x,y,z$ positive integers the form of $\frac{x^2+y^2+z^2}{xy+yz+zx}$ can't be equal to 3. [duplicate]

Because it is positive integers so I can multiply both side by $(xy+yz+zx)$ I have tried to use completing square, like below $(3x-3y)^2+(3y-3z)^2+(3z-3x)^2=12x^2+12y^2+12z^2$ But in above form, it ...
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### Help with the indefinite integral $\int \frac{dx}{2x^4 + 3x^2 + 5}$

I start by rewriting the denominator, $2x^4+3x^2+5$, as a squared term plus a constant. To do this, we notice that the first two terms already have a common factor of $2x^2$. We can complete the ...
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### Can the substraction of quadratic form ($(x-b)^T A(x-b)$) of two functions be written as another quadratic form plus constant

A function $f: \mathbb R^d \rightarrow \mathbb R$ is said to have quadratic form iff it can be written as $f(x)=(x-b)^T A(x-b)$ for some $A\in \mathbb R^{d\times d}$ and $b \in \mathbb R^d$. If I have ...
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### Isolating x in a quadratic expression by completing the square

I came across some study notes regarding quadratic expressions and there is a solution that I am having a hard time understanding. In the study notes it is stated: "The variable x in the ...
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1 vote
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### How do you write $4(x +1)^2 + 1$ in the form $(ax +b)^2 + c\;$?

We can write $4x^2+8x+5$ in the form $a(x+b)^2+c$ as $4(x+1)^2+1$. However, the question I am doing asks me to write it in the form $(ax+b)^2+c$. How do I change it to that form?
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### Show path of the particle is hyperbolic

The equation for the path of the particle in cartesian coordinates is $$(1-e^2)x^2+y^2+2exp=p^2$$ where $p$ and $e$ are constant. Under the condition that $0<e<1$, I have shown that this ...
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1 vote
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### Converting $ax^3 + bx^2 +cx + d$ to $a(x-j)^3 +k$

We're all familiar with the vertex form of a quadratic function,$$a(x-p)^2 +q$$ where $(-p,q)$ represent the coordinates of the maximum or minimum point of the parabola. This is achieved by performing ...
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### What are the application of "completing the square" method?

I am trying to convince my students that "completing the square" method is widely used in mathematics. Here is my effort, could you add other applications to my list? Finding the maximum or minimum ...
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