Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [completing-the-square]

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

To illustrate with a simple example, to complete the square for an expression such as $x^2+6x-1$ means to add another term (or terms) in such a way that the result is a perfect square:

$$(x^2+6x-1)+10=(x+3)^2\ .$$

When applied to quadratic expressions like the one above, completing the square helps us see geometric information about the graph of the expression. For example, the graph of $y = x^2+6x-1$ is a parabola that has its vertex at the coordinates $(-3,-10)$, which we can see in the re-expression of the equation as $y = (x+3)^2-10$. In general, the formula for completing the square of a quadratic polynomial looks like

$$x^2+bx+c \;\;=\;\; \left(x+ \frac{b}{2} \right)^2 +c - \frac{b^2}{4}$$

This concept can be extended to functions of more than one variable.

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I think I can complete the square of any quadratic, is it true? (Any reason to ever use Quad. Formula?)

I was taught that you could only complete the square of a quadratic if the coefficient on the $x^2$ term is 1. However, playing a little bit with other quadratics, I've found that it's just not true. Based on the CTS algorithm, you just need to…
jeremy radcliff
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If $3x^2 -2x+7=0$ then $\left(x-\frac{1}{3}\right)^2 =$?

If $\ 3x^{2}-2x+7=0$ then $$\left(x-\frac{1}{3}\right)^2 =\text{?} $$ I am so confused. It is a self taught algebra book. The answer is: $ \large -\frac{20}{9}$ but I don't know how it was derived. Please explain. Thanks for everyone who commented!…
Nicki
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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) }3 \space \space \space \space \space\text{D) }4…
Hussain-Alqatari
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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 generally, conic sections) in $\mathbb{R}^2$, it's…
πr8
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Completing the square with negative x coefficients

I know how to complete the square with positive $x$ coefficients but how do you complete the square with negative $x$ coefficients? For example: \begin{align*} f(x) & = x^2 + 6x + 11 \\ & = (x^2 + 6x) + 11 \\ & = (x^2 + 6x + \mathbf{9}) + 11 -…
Kot
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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 help
Math_reald
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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 square by taking half of the coefficient of our $x^2$…
Meharaj hossain Arman
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Completing the square to show that $4x^2 - 5y^2 + 24y = 16$ is a hyperbola

I was wondering how would I complete the square for this particular hyperbola?$$4x^2 - 5y^2 + 24y = 16$$ I tried this technique but to no avail: \begin{align*} 4x^2 - 5(y^2 + \frac{24}{5}y) & = 16\\ 4x^2 - 5(y + \frac{12}{5})^2 & = 16 +…
Person
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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 and vectors with all real entries. All boldface…
Clarinetist
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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 $$ \int_{-\infty}^\infty dx~\exp[-ax^2]\exp[bx] $$ where they completed the square and got…
Illari
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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 \\ & = a\left(x^2+{b\over a}x+{b^2\over 4a^2}-{b^2\over…
Decaf-Math
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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, it cuts $b$ in half, and moves it under the $x^2$…
Obinna Nwakwue
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Completing the square of $(x+a)(x+b)$

The problem is simple, to complete the square of $(x+a)(x+b)$. My calculations yield $$\left(x+\frac{a+b}{2}\right)^2-\frac{(a+b)^2}{4}+ab,$$ But the textbook's answer is different ("problem 361", at the bottom of the…
CopperKettle
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Complete the squares to find the center and radius of the circle

I've had trouble on this one problem for a couple days. Complete the square on the X and Y terms to find the center and radius of the circle. $x^2+2x+y^2-4y=-4\:\:$
LizHenson
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Non-cyclic inequality with fractions

I need help with following inequality problem: Let $a,b,c$ are positive real numbers $\mathbb{R}^+$. Prove that $$ \frac{2}{a^2}+\frac{5}{b^2}+\frac{45}{c^2}>\frac{16}{(a+b)^2}+\frac{24}{(b+c)^2}+\frac{48}{(a+c)^2}. $$ I was unable to find any…
Oliver Bukovianský
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