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Questions tagged [completing-the-square]

Questions on the algebraic operation of completing the square.

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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}+\...
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How to complete the square with 3 terms

I was wondering what the general approach should be when it comes to completing the square with three terms (three terms within the braces being squared). eg. Write the following function as a sum of ...
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Completing the square

$2x^2 - 6x - 5$ My workings are $2 ( x^2 - 3x + (3/2)^2 - (3/2)^2 - 2.5 ) = 0$ $ 2 ( x - 1.5)^2 - 4.75 = 0$ $ (x-1.5)^2 = 2.375 $ From here I go on to find X which is not the correct answer .. ...
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Why does $\left(\frac b2\right)^2$ “geometrically complete the square?”

I was just reading this MathisFun article on completing the square. It states that geometry can help complete the square. It starts off with a square and a rectangle (pictures come from link): Then, ...
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tricky completing the square

$$\frac{4k_{0}p_{1}+p_{0}\left ( k_{0}^{2}-2k_{0}k_{1}+k_{1}^{2} \right )}{\left ( k_{0}+k_{1} \right )^{2}p_{0}}$$ I need to show that this is equal to $1$ but for my life I can't figure how to ...
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State the range of this

How do I state the range of the following equation: $$8x-4x^2$$ using $$c-\frac{b^2}{4a}$$ I'm again unsure of what $a$, $b$, and $c$ are.
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A B and C in Completing the square?

When I complete the square, and I get this for example: $$-(x+3)^2 - 13$$ What would $A, B$ and $C$ be? For example, when using $C - B^2 / 4A$ Thanks!
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Completing the square (and variants thereof)

When dealing with quadratics, completing the square is ubiquitous, and I can summarise my interpretation of it as the formula: $$x^2-2ax=(x-a)^2-a^2$$ Likewise, when working with circles (and, more ...
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Converting to vertex form, where coefficient of $w^2$ cannot be factored out

I need help on converting this to vertex form: $$12w^2 + 13w + 3$$ I have tried finding examples online, but every time I find an example where $x^2$ has a coefficient, it is always able to be ...
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Method of completing squares with 3 variables

I want to use the method "completing squares" for this term: $x^2-2xy +y^2+z^2*a+2xz-2yz$ The result should be $(x-y+z)^2 +(a-1)*z^3$ Is there a "recipe" behind how to do this? Hope someone could ...
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Finding the inverse laplace of this function: $ F(s)= \frac{s+8}{s^{2}+4s+5}$

Im trying to find the inverse laplace of : $ F(s)= \frac{s+8}{s^{2}+4s+5}$ I reached the following: $$ F(s)= \frac{s}{(s+2)^{2}+1} + 8 \times \frac{1}{(s+2)^{2}+1}$$ Now i have the 2nd term in the ...
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246 views

Completing squares - matrices

I am familiar with completing squares in R and I am familiar with matrix-equations - but I dont have an idea, how to do completing squares with matrices. $\frac{1}{2}X^TCX+b^TX+A$ The solution is $\...
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1answer
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Computing a double integral

Consider the double integral $$\int dp \int dp' \exp[-\frac{1}{2}p^2 + Kpq] \exp[-\frac{1}{2}p'^2 + Kpp' + Kp'q'],$$ where $q$ and $q'$ are constants. How should one solve such an integral? I tried ...