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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7
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3answers
205 views

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

If $\int_{0}^{\infty}\frac{dx}{1+x^2+x^4}=\frac{\pi \sqrt{n}}{2n}$, then $n=$ $\text{A) }1 \space \space \space \space \space\text{B) }2 \space \space \space \space \space\text{C) }3 \space \space \...
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1answer
49 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 ...
1
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3answers
196 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?
2
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3answers
149 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 ...
0
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0answers
44 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+...
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2answers
50 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} $
2
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2answers
85 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 $$ \...
2
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0answers
44 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 ...
2
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1answer
96 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 \...
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3answers
93 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.
2
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2answers
109 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, $C\in\mathbb R^{n\times n}$...
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0answers
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 ...
0
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1answer
55 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 ...
0
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1answer
165 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^...
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2answers
42 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 ...
1
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1answer
157 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 $ ...
1
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3answers
476 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 ...
2
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0answers
64 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 ...
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2answers
219 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 ...
2
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0answers
216 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|...
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1answer
140 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 $z'$??
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1answer
88 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 ...
2
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4answers
203 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?
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5answers
628 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.
0
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1answer
20 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.
2
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6answers
105 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)=...
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3answers
27 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 ...
1
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2answers
36 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 ...
0
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1answer
100 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 $...
1
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4answers
58 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 ...
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2answers
152 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 ...
3
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1answer
274 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 ...
0
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2answers
118 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 ...
0
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2answers
151 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 $a(x+b)^2+c$.
4
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0answers
2k views

completing the square in a gaussian integral

I'm trying derive this integral $$ I = \int_{-\infty}^\infty dx~\exp[-ax^2 + ikx] $$ I was following someone else's work for a similar integral of $$ \int_{-\infty}^\infty dx~\exp[-ax^2]\exp[bx] $...
0
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1answer
53 views

Completing square in hyperbolic equation

I tried completing the square for the following hyperbola: $x^2+y^2-3xy+2x+3y+2=0$ by sending $3xy+2$ to the other side and finally stuck here: $(x+1)^2 +2(y+\frac{3}{4})^2 = \frac{1}{8} -3xy$ I am ...
1
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1answer
1k views

Completing the Square with an A greater than 1

I need some assistance with a specific problem where the equation given is $-3x^2-3x+9=0$ I have divided everything by $-3$ to get $x^2+x-3=0$ Then I move the $3$ to the other side $x^2+x=3$ ...
0
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2answers
54 views

A question about a step in completing the square to prove the quadratic formula

$$\left(x+\dfrac{b}{2a}\right)^2 = \dfrac{-4ac+b^2}{4a^2}$$ goes to : $$\left(x+\dfrac{b}{2a}\right)^2 = \dfrac{b^2-4ac}{4a^2}$$ I don't understand how the signs changed in going from -4ac+b^2 to b^...
1
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1answer
40 views

Basic multivariate Factorization type question

I came across this polynomial $$ x^2 - 2kxy + ky^2 +d; \, k>0 $$ in some of my work and was wondering if there was a trick to coercing/factoring it into a polynomial of the form $$ (x-x_0)^2-(y-y_0)...
1
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2answers
114 views

completing square for a circle

In the following question: I don't understand how we can get from the original equation to the final equation using completing the square. Any thoughts as how to get to the final equation?
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3answers
56 views

Completing the square of $x^2 - mx = 1$ is not giving me the right answer.

This is my attempt $$ \begin{align} x^2 - mx &= 1 \\ x^2 - mx - 1 &= 0 \\ \left(x^2 - mx + \frac{m^2}{4} - \frac{m^2}{4}\right) - 1 &= 0 \\ \left(x^2 - mx + \frac{m^2}{4}\...
0
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1answer
38 views

Completing the Square (Bivariate)

I am running a bivariate regression with the formula $$\hat{Z} = b_0+b_1X+b_2Y+b_3X^2+b_4Y^2+b_5XY$$ I'm easily able to obtain the 6 coefficients this way. However, I want to reparameterize the ...
1
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2answers
129 views

When to factor to multiple terms and when to complete the square?

I've been working through some tutorials that show how to factorised in multiple ways, e.g. $2x^2 + 11x +12$ Can be factorised to: $(2x + 3)(x + 4)$ ... and to: $2[(x + \frac{11}{4})^2 - \frac{25}...
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1answer
505 views

Conic section General form to Standard form Hyperbola

Hi I'm attempting to change $9x^2-18x-y^2-8y-88=0$ to standard form. Here is what I've done: $$(9x^2 - 18x -1) - (y^2 + 8y +16) = 88+9+16$$ $$(3x-3)^2 - (y+4)^2 = 113$$ $$(3x-3)^2/113 - (y+4)^2/113 = ...
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4answers
2k views

Why does completing the square work?

I am currently learning about quadratics in high school and we've just done completing the square. Now I understand how to complete the square, I just don't understand why we can complete the square. ...
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2answers
78 views

Solve quadratic related problem without brute force/trial and error or quadratic formula.

At least I need to explain to an 11 year old One square is cut out of another. Side lengths of each square are whole numbers less than 25. Remaining area of larger is 57. What is perimeter of ...
3
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5answers
520 views

$Why$ is the axis of symmetry of a parabola $-{b\over 2a}$ and ${not}$ ${b\over 2a}$?

I'm working on a lesson plan for my students regarding completing the square for a parabola, and I've done the following: $$\begin{align}ax^2+bx+c &= a\left(x^2+{b\over a}x\right) + c \\ & =...
0
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2answers
69 views

Completing the square of $(m^2 + n^2)$

I'm attempting to complete the square of $m^2 + n^2$, How would I do this? I am not understanding, as most resources refer to a polynomial with $x$ as it's variable and every term is in terms of $x$. ...
2
votes
2answers
2k views

help solve simultaneous equation

I need to solve the system $$6x+y=3\tag 1 $$ $$x^2+y^2=16\tag 2$$ So first, I rearrange $(1)$ to $y=3-6x$ and substitute that into $(2)$. I get $$x^2+(3-6x)^2 = 16$$ which is $$x^2 + 36x^2-36x+9-...
1
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1answer
51 views

Question on completing the square obtaining the form $a(x+p)^2+q$

From the last term in completing the square from the quadratic $ax^2+bx+c$, I was just wondering how $$-\left(\frac{b}{2a}\right)^2+c = \left(c-\frac{b^2}{4a}\right)$$ I would have gotten $$-\left(\...