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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A Complete Guide to Completing The Square

I've noticed recently that many people on this site are not so familiar with the method of completing the square with quadratic equations. So, this question and my answer is a full explanation on ...
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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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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 ...
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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 ...
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hey !! i did part a but i am stuck with part b..Can anyone help me out? [closed]

(a) Write the quadratic expression $ax^2+bx+c$ in the form $a(x+h)^2+k$ for suitable values of $h$ and $k$ in terms of $a$, $b$, and $c$. (b) When does the equation $a(x+h)^2+k=0$ have solutions? ...
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44 views

How can one show that $\sqrt{5+2\sqrt{6}}=\sqrt{3}+\sqrt{2}$ is true? [closed]

I was searching for some knowledge about algebra and found something interesting I found a an equation of 4th order that looks like this $x^4-10x^2+1 = 0$ in the article that I was reading, I found ...
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52 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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Calculation of multivariate gaussian integral

I'm currently working on understanding the paper "Autoregressive Point Processes as Latent State-Space Models: A Moment-Closure Approach to Fluctuations and Autocorrelations" by Rule & Sanguinetti....
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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 ...
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61 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 $ ...
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103 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 ...
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58 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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59 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 ...
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125 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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60 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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80 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 ...
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119 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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111 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.
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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.
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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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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 ...
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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 ...
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1answer
81 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 $...
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54 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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129 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 ...
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172 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 ...
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109 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 ...
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116 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 $a(x+b)^2+c$.
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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] $...
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1answer
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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 ...
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1answer
166 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 $x^2+x=3$ ...
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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^...
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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)...
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100 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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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}\...
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1answer
29 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 ...
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38 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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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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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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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 ...
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$Why$ 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 \\ & =...
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66 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 $x$. ...
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1k 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-...
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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(\...
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How to get $\frac{2}{3}(3x-5)^{2}+\frac{19}{3}$ from this expression $6x^{2}-20x+23$.

I'm trying to figure out how to get $\frac{2}{3}(3x-5)^{2}+\frac{19}{3}$ from this expression $6x^{2}-20x+23$. Hints?
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Trig Substitution Help

I need to integrate the following using trigonometric substitution. I also know that I need to do the following by completing the square in the denominator, but I can't seem to figure it out. $$\int \...
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667 views

Find inverse of a quadratic polynomial by 'completing the square'

I have been asked to find the inverse of an equation that has the form $y=ax^2 + by -c$ EDIT: Which is $y=4x^2+ 8x -3$ in the graph below Using a graphing calculator, and trial and error, I can find ...
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105 views

Solve $y = \frac{x^6}{6} + \frac{1}{16x^4}$ by completing a perfect square, with a given domain. [closed]

Problem: $$y = \frac{x^6}{6} + \frac{1}{16x^4} \text{ for } 4 \le x \le 25$$ Solve by completing the square and use regular anti-derivatives. Not even sure where to begin. I'm taking online classes ...
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378 views

Is it really necessary to learn how to write a quadratic in standard to vertex form?

How crucial is this skill or form of writing and polynomial function? Can't we just always use the $-b/2a$ trick for the $x$ intercept and just plug it back into the function to find the $y$?
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Find all real numbers x, y, z, u and v in $\sqrt{x}+\sqrt{y}+2\sqrt{z-2}+\sqrt{u}+\sqrt{v}=x+y+z+u+v$

And thanks in advance for your answers. Sorry if the text is badly formatted, I'm new here. Anyway, here is the question: Find all real numbers x, y, z, u and v in $\sqrt{x}+\sqrt{y}+2\sqrt{z-2}+\...