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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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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Completing the squares

If I have the following equation: $$f = 2\gamma_2 \sigma_{2,1}^2 a^2 + 4 \gamma_2\sigma_{2,1}\sigma_{2,2}ax_2 + 2\gamma_2 \sigma_{2,2}^2x_2^2$$ I can write it as: $$f = 2\gamma_2(\sigma_{2,1}a + \sigma_{2,2}x_2)^2$$ But now I have: $$g = 2\gamma_1…
Pietair
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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!
Alex
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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 the square", which I know but I can't understand…
WorldGov
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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 problem and couldn't figure out why this happens,…
J.Doe
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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}\right) - \frac{m^2}{4} - 1 &= 0 \\ \left(x^2 - mx +…
pragMATHiC
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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 $z'$??
lara sandy
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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.
abee 99
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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 $x$ will give the desired powers for $u$. Subbing in…
spyr03
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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 $a(x+b)^2+c$.
Liam Michel
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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^2-4ac. I am a beginner at maths ; I have tried to…
Bahar
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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 $x$. Edit: I'm trying to do this in a proof. I'm…
Howard P
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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 .. Can anyone help me on where I have gone wrong ? Thanks…
user307640
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Completing the square

Can someone explain what happens from step 6 forward, when solving 3x^2 – 12x – 7 = 0 by completing the square. How does radical of 19/3 turn into radical 57/ 3?
Rndpbs
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