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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 ...
Lim Zhao Sen's user avatar
4 votes
3 answers
222 views

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 ...
Meharaj hossain Arman's user avatar
0 votes
0 answers
28 views

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 ...
Interception's user avatar
0 votes
0 answers
51 views

Completing the square under a square root

Trying to work through a problem that requires us to complete the square, then use a sinh substitution...but I need to start by remembering back to math from many years ago. $$\int\sqrt{2x^2+3x+4} \ ...
usuallyBadAtMath's user avatar
0 votes
2 answers
76 views

Completing the square: How does $5x^2−4Nx+2N^2$ get to this?

I've been stuck on this for the past few hours. For context, you can refer to this thread, and in particular this answer which makes sense, but skips over all the steps. How does $5x^2-4Nx+2N^2=$ ...
Rene's user avatar
  • 103
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0 answers
25 views

Find the maximum or minimum y-value of the quadratic function by completing the square.

I have a question that reads: "Find the Minimum or Maximum (state which it is) y-value of the following quadratic function by completing the square..." Followed by the equation: $y = \dfrac{...
Cyrus Martin's user avatar
0 votes
1 answer
68 views

Completing the square at GCSE level 9/9

I am reading page 42 of a revision booklet from 2015 called GCSE AQA Mathematics Higher Level, Complete Revision Guide and Practice, by CGP.. The topic is completing the square. Below is a copy of the ...
securityauditor's user avatar
-2 votes
1 answer
74 views

Completing the square with this polynomial? [closed]

$$\int \dfrac{1}{\sqrt{4x^2+4x+17}}\,dx$$ So far I have only worked with polynomials of the type $x^2$, not $Ax^2$ what should I do to complete the square in this case?
SirMrpirateroberts's user avatar
2 votes
3 answers
276 views

Whats wrong with this integral?

$$\int \frac{(x+7)dx}{x^2+2x+5}$$ completing the square we get: $$(x^2+2x+1)+5-1$$ $$(x+1)^2+4$$ then: $$U = x + 1$$ $$a = 2$$ hence: $$\int \frac{(U+6)dU}{U^2+a^2}$$ using trig substitution $U = a\...
SirMrpirateroberts's user avatar
2 votes
1 answer
61 views

What did I get wrong in this integral?

$$\int \frac{(x+1)dx}{\sqrt{2x-x^2}}$$ completing the square: $$1-(x^2-2x+1)$$ $$1-(x+1)^2$$ $$a^2-U^2$$ we have: $$U = (x+1) $$ $$a = 1$$ we have: $$\int \frac{UdU}{\sqrt{a^2-U^2}}$$ let: $ U= a\sin\...
SirMrpirateroberts's user avatar
2 votes
1 answer
59 views

How to complete the square of this polynomia

let: $$\int \frac{dx}{x^2+4x+5}$$ I am trying to complete the square in order to use trig substitution but I failed I was trying: $$(x^2+4x+1)+5-1$$ $$(x^2+4x+1)+4$$
SirMrpirateroberts's user avatar
0 votes
1 answer
228 views

Completing the square in a Gaussian distribution [closed]

Let the distribution $\displaystyle p(\theta | x) = L(\theta | x)p(\theta) = p_{1} = \exp[-\frac{1}{2} ((\frac{\theta - x}{\sigma})^{2} + (\frac{\theta - \mu_{0}}{\sigma_{0}})^{2})]$ where $L$ is the ...
Mathematicing's user avatar
0 votes
1 answer
252 views

Normal distribution for $Z=X+Y$ where $X,Y$ are both normally distributed

I need to find the probability distribution for $Z = X+Y$ where $X \sim \mathcal{N}(x_0,\sigma_x^2)$ and $Y \sim \mathcal{N}(y_0,\sigma_y^2)$. $X$ and $Y$ are independent. In order to do this, we use ...
Tanamas's user avatar
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1 answer
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Diagonalization calculator of definite quadratic forms using the "completing the square" method

In this question: Calculating the maximum value of a quadratic polynomial on several variables with some restrictions, I asked about finding the maximum value of a certain quadratic polynomial with ...
user302934's user avatar
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1 vote
1 answer
76 views

Writing $axy + bx + cy + d$ in the form $a(x-\alpha)(y-\beta)+\gamma$

I came across a problem in which we needed to rewrite an equation of the form: $$axy + bx + cy + d$$ in the form $$a(x-\alpha)(y-\beta)+\gamma$$ where $a,b,c,d$ are known constants and $\alpha,\beta,\...
FD_bfa's user avatar
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1 answer
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Completing squares

We want to find when a quadratic form depending on a parameter $\alpha$ on $\mathbb{R}^3$ is an inner product. It is quite obvious and quick to deal with the problem by using Sylvester, BUT we want to ...
Sam Skywalker's user avatar
7 votes
3 answers
367 views

If $\int_{0}^{\infty}\frac{dx}{1+x^2+x^4}=\frac{\pi \sqrt{n}}{2n}$, then $n=$

$$\text{If }\int_{0}^{\infty}\frac{dx}{1+x^2+x^4}=\frac{\pi \sqrt{n}}{2n},\text{then } n=$$ $$$$ $$\text{A) }1 \space \space \space \space \space\text{B) }2 \space \space \space \space \space\text{C) }...
Hussain-Alqatari's user avatar
0 votes
1 answer
174 views

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 ...
Austin737's user avatar
  • 111
1 vote
3 answers
228 views

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?
daisie's user avatar
  • 37
2 votes
3 answers
241 views

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 ...
In the blind's user avatar
0 votes
0 answers
100 views

Complete the square of a multivariable function

I have a quadratic form which I have mostly completed the square but I am stuck at the final stage. Currently it is in the form: $\frac{1}{2}\left(x+4y\right)^2+4\left(y+z\right)^2+\frac{1}{2}\left(x+...
PythonIsBae's user avatar
1 vote
2 answers
98 views

How to find the minimum of $xy$ from the equation $2x+y+6=xy$ , $x,y>0$

I have tried doing factoring and completing the square and also AM-GM $(2-y)(x-1)=-8$ $xy=2x+y+6≥\sqrt{12xy} $
Tom Xia's user avatar
  • 151
2 votes
2 answers
88 views

Combining and simplifying equations

I've come across a derivation in a journal article where one of the steps is to substitute one expression into another, but I cannot simplify to their solution. The Expressions are: Expression 1 $$ \...
user18483's user avatar
2 votes
0 answers
57 views

Tough integration problem - complete the square, domain, absolute value, plus/minus.

I almost made it through a really tough integration problem, but I ended up with some extraneous +/-'s that I could only eliminate by graphing the solution. Symbolab seems to have the correct answer ...
user3925803's user avatar
2 votes
1 answer
104 views

Evaluating $\frac{1}{(4\pi t)^{\frac{d}{2}}} \int_{\mathbb{R}^{n}} e^{-\frac{|x-y|^2}{4t}}e^{-y^2}dy$

I wish to find the function $u(x,t)$ which solves the following PDE: $$\begin{cases} \partial_{t}u - \Delta u = 0, \; t \in \mathbb{R}_{+}, \; x \in \mathbb{R}^{d} \\ u(0,x) = e^{-x^{2}}, \; x\in \...
ozera ozera's user avatar
0 votes
3 answers
304 views

How to factorize $2x^2 + 2x + 1$?

How to factorize $2x^2 + 2x + 1 = 0$ into $2(x + 0.5)^2 + 0.5$ ? What is the process behind this factorization? Edit: Thanks guys, have updated this post. Updated the tag to complete the square.
Dugong98's user avatar
  • 137
3 votes
2 answers
253 views

Frobenius norm: completing squares and minimizing

I would like to minimize the following quantity: $Q = \left\lVert{X - C}\right\rVert^2_F + a\left\lVert{X - I}\right\rVert^2_F$ Where $X\in\mathbb R^{n\times n}$ is unknown, $C\in\mathbb R^{n\times n}$...
cgss's user avatar
  • 1,098
0 votes
0 answers
26 views

Completing the square of complicated expression (answer available)

I would like to rewrite the following expression $$ \begin{align} \frac{1}{\sigma_{\theta}^{2}}\sum_{i=1}^{I}(\theta_{i}-\mu)^{2} + \frac{1}{\sigma_{0}^{2}}(\mu-\mu_{0})^{2} \end{align} $$ as a square ...
SimpleProgrammer 's user avatar
0 votes
1 answer
143 views

How to convert 3 variable quadratic form into a sum of squares?

How do I convert the 3-variable quadratic form q(x₁ x₂ x₃) = row vector [x₁ x₂ x₃] multiplied by [a₁₁ a₁₂ a₁₃ a₂₁ a₂₂ a₂₃ a₃₁ a₃₂ a₃₃]₃ₓ₃ matrix multiplied by column vector [x₁ x₂ x₃] into the ...
Francis Scott's user avatar
0 votes
1 answer
390 views

Lagrange Method to complete the square

I'm having a really hard time solving this equation: $q(x_1,x_2,x_3,x_4)=4x_1x_4 + 2x_2x_3$ I tried many different approaches and got to $(4x_1^2+4x_1x_4+x_4^2-4x_1^2-x_4^2)+2x_2x_3= (2x_1+x_4)^2-4x_1^...
Milo's user avatar
  • 3
0 votes
2 answers
47 views

Can't find error in completing the square

Now matter what I do I can't seem to find the error in my completion of the square... It's probably something obvious but I'm running out of ideas at this point. I'm trying to complete : $ 3 x^{2}-2 ...
A.D's user avatar
  • 103
1 vote
1 answer
330 views

Simplify integral of exponential functions using the method of completing the square

How can I simplify $ 1 - \frac{1}{\sqrt{2\pi}\sigma}\int^{\infty}_{-\infty}e^{-kw}e^{\frac{-(w-\mu)^{2}}{2\sigma^{2}}}dw$ using the method of completing the square? I know that the answer must be $ ...
Arnaud Wolff's user avatar
1 vote
3 answers
1k views

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 ...
iciqle's user avatar
  • 35
2 votes
0 answers
100 views

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 ...
Display Name's user avatar
  • 2,715
0 votes
2 answers
401 views

How are these related? completing the square vs. graphing a quadratic equation

While searching to learn about complex numbers on the Internet, I was referred also to quadratic equations. Several graphic examples showed how "completing the square" uses a quadratic equation to ...
Peter Buxton's user avatar
2 votes
0 answers
352 views

Completing the square with complex numbers

Dear MSE-community! I have begun on a journey through Gamelin's Complex Analysis. In the first chapter is an exercise described in the Math.SE question "Show that the set $z$ satisfying $|z−z_0|=\rho|...
iaenstrom's user avatar
  • 399
0 votes
1 answer
231 views

Completing the square of: $y^2-xz$

How can I complete the square of the following monomial: $y^2-xz$ to obtain the sum of 3 squares of the form: $y'^2-z'^2-x'^2$. Any suggestions for finding $x',y'$ and $z'$??
lara sandy's user avatar
0 votes
1 answer
107 views

Completing the Square with a quartic polynomial

Find all integers $n$ for which $81\frac{n^4}{4}-2017n^2+81$ is a prime. I know completing the square helps with this problem, but I'm not how completing the square is going to get me to the right ...
Dave Ezarik's user avatar
2 votes
4 answers
249 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?
Jenny B's user avatar
  • 367
0 votes
5 answers
3k views

How to solve $9x^2+6x-5$ by "completing the square" method so that we can get the result as $(3x+1)^2 +6$? [closed]

I am having trouble getting the answer using "completing the square" method. Please explain all the steps.
ri_zen's user avatar
  • 13
0 votes
1 answer
22 views

Quadratic Function: possible value of c

I have $f(x)=x^2-2bx+c$ If the minimum value of the function is 6, what is the possible value of c. I tried$$f(x)=(x-b)^2-b^2+c$$ $$b^2=c-6$$ I couldn't solve for the value of c.
abee 99's user avatar
  • 37
2 votes
6 answers
150 views

How to factorize $2x^2-9x+9$ by completing the square?

I know that $x^2-bx+c=(x-k)^2=x^2-2kx+k^2$ if it is a complete square. If not we create one by adding and subtracting $\left(\frac{b}{2}\right)^2$ I tried $$2\left(x^2-\frac{9}{2}x+\frac{9}{2}\right)=...
Hills's user avatar
  • 155
1 vote
3 answers
29 views

How does this transformation happen?

$$ ax^2+2hxy+by^2 = a\left(x+ \frac{h}{a}y \right)^2 + \frac{ab-h^2}{a}y^2 $$ How do I go from the equation on the LHS to the equation on the right hand side? My study material mentions "completing ...
WorldGov's user avatar
  • 957
1 vote
2 answers
48 views

Why doesn't adding the equations in this system of equations find the solution?

I have two equations: $x^2 - y = 0$ and $y^2 - x = 0$. Adding them gives $x^2 - x + y^2 - y = 0$, and completing the square results in $(x - 1/2)^2 + (y-1/2)^2 = 1/2$. This suggests that there is an ...
user124384's user avatar
0 votes
1 answer
130 views

Rewrite (a quartic) equation as a square

Given the equation $$\frac{u^4}{3} - 2u^3 + \frac{23}{3}u^2 - 6u + 8$$ I want to rewrite it in the form $x^2 + 7$. As it is a quartic, I started by letting $x = au^2 + bu + c$, since squaring this $...
spyr03's user avatar
  • 1,024
1 vote
4 answers
63 views

Is this a correct derivation of completing the square?

$x^2 + bx$ $=x^2 + bx + c - c$ $=(x + k)^2 - c$ $=x^2 + 2kx + (k^2 - c) = x^2 + bx + 0$ This implies: $2k = b$, so $k = b/2$, and: $k^2 - c = 0$, or $k^2 = c$, or $(b/2)^2 = c$ So to complete ...
user51819's user avatar
  • 1,161
1 vote
2 answers
175 views

Can't find error at completing the square

I am desperatly looking for the mistake I did when completing the square. I have a function $f(x)=-4.905x^2+5x+6$ Nothing special. So when I was trying to find the peak of the curve I ran into a ...
J.Doe's user avatar
  • 526
4 votes
1 answer
531 views

Completing the square in $N$ dimensions

This is very important for Bayesian methods in statistics, but I haven't been able to find a reference which specifically touches on my situation. Assume all matrices and vectors below are matrices ...
Clarinetist's user avatar
  • 19.6k
0 votes
2 answers
160 views

Completion square method quadratic form

$-6xy-6xz+3y^2+6yz-2z^2$ I've already tried to factor out some variables, but I am always left with 3 variables again after my transformation. I've tried $(a+b)^2$ and $(a+b+c)^2$, I guess I need ...
Alice's user avatar
  • 109
0 votes
2 answers
302 views

completing the square with a coefficient more than 1.

I've tried to solve it, is this right? $$2x^2+6x+35=0$$ $$2(x^2+6x)+35$$ $$2(x+3)^2+35-9=0$$ $$2(x+3)^2=26=0$$ I was told to write it in the form $a(x+b)^2+c$.
Liam Michel's user avatar