Completing The Square?

Question

Can anyone give me a detailed explanation with examples but simplistic and easy to follow on how to do 'Completing The Square'?

Answer 1

Sure, I can try. Let's try completing the square for this quadratic.

Example 1:

y = x^2 + 4x + 18

The trick to completing the square is to note the coefficient (the number in front of) x. What you want to do is take "half squared" of this coefficient. By "half squared" I mean take half of the value (or, multiply it by 1/2), and then square the result.

In our example, half of 4 is 2, squared is 4. So 4 will be our magic number.

Now, what we're going to do is ADD and then SUBTRACT 4. This doesn't change *anything* because adding and subtracting a number is equivalent to adding ZERO, which in turn does nothing. We're going to INSERT a 4 before the 18, and insert a -4 after the 18.

y = x^2 + 4x + 4 + 18 - 4

Now, our first three terms form a square trinomial, and becomes

y = (x + 2)^2 + 18 - 4

{Notice how I strictly took half of the coefficient of x this time; this is what goes in the brackets after making it a square term.}

Simplify a bit more, and we have

y = (x + 2)^2 + 14, and now we have completed the square!

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Example #2:
y = x^2 + 7x + 14

Again, we determine the number we need to add and subtract. Remember, that it is "half squared" of the coefficient of x.
Half of 7 is 7/2, squared is 49/4. Recall that taking half of 7 is equivalent to multiplying by (1/2), and recall that squaring a fraction is equivalent to squaring top and bottom, hence 49/4.

Now, ADD and SUBTRACT this value.

y = x^2 + 7x + 49/4 + 14 - 49/4
y = (x + 7/2)^2 + 14 - 49/4

Merge the last two fractions by putting them each over 4.

y = (x + 7/2)^2 + 56/4 - 49/4
y = (x + 7/2)^2 + 7/4

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Example #3:
y = -2x^2 + 12x - 11

Key thing to note here: This example differs from the rest because the x^2 term has a -2 in front. That means we have to take a different first step.

Your first step is to FACTOR out -2 from the first two terms.

y = -2(x^2 - 6x) - 11

Now, we need "half squared" of -6. Half of -6 is -3, squared is 9. So

y = -2(x^2 - 6x + 9) - 11 + ?

Remember how we normally simply added the term and then subtracted it? We CANNOT do that in this case; putting -9 for the question mark would be incorrect.

Let's look at what we did; we added a 9 INSIDE the brackets where there's a lingering -2. Therefore, what we REALLY added was -2(9), or -18. To offset this, we have to ADD 2(9), or 18!

y = -2(x^2 - 6x + 9) - 11 + 18

Now, solve as normal.

y = -2(x - 3)^2 + 7

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Example 4:

y = 3x^2 + 7x + 1

Factor out 3 from the first two terms.

y = 3(x^2 + (7/3)x) + 1

Now, add "half squared" of 7/3. (1/2)(7/3) = 7/6, squared is
49/36. Therefore

y = 3(x^2 + (7/3)x + 49/36) + 1 + ?

Remember that we really added 3(49/36), due to adding it in the brackets with the 3. So we SUBTRACT 3(49/36), which is the same as subtracting 49/12.

y = 3(x^2 + (7/3)x + 49/36) + 1 - 49/12

y = 3(x + 7/6)^2 + 1 - 49/12

Put the last two terms under a common denominator, then merge into one.

y = 3(x + 7/6)^2 + 12/12 - 49/12
y = 3(x + 7/6)^2 - 37/12

Answer 2

Completing the square

x² + 8x + 4 = 0

x² + 8x + 4 - 4 = 0 - 4

x² + 8x = - 4

the 8x divide the 8 by 2 = 4 and square the 4² = 16, This will get a perfect squae trinomial

x² + 8x + 16 = - 4 + 16

(x + 4 )(x + 4) = 12

√(x + 4)² = √12

x + 4 = ± √4 √3

x + 4 = ± 2 √3

x + 4 - 4 = - 4 ± 2 √3

x = - 4 ± 2 √3

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Click on the URL below for additional information concerning completing the square.

www.purplemath.com/modules/solvquad3.htm

Answer 3

consider quadratic equation a(x^2 )+ b(x) + c = 0

in order to complete the square do bear in mind that we can only make adjustments for the constant term 'c'. Co-efficient of x^2 which is 'a' has to be a perfect square.Find out its root, say root_a

so making sure 'a' is a perfect square by itself ( most cases it is 1), break up 'b' as 2*root_a*k .
Example 4(x^2) + 18(x) +56 = 0
a = 4 , root_a = 2, b= 18 , c=56
18 = 2 * 2 *k
=> k = 18/4= 9/2 = 4.5

Next step is to square k , i.e (4.5)^2

re-write the equation as
[4(x^2) + 2 * 2 * (4.5) (x) + (4.5)^2 ] + (56 - (4.5)^2) = 0

what we did in the previous step is to separate the k^2 term in the constant c and write the remainder outside the brackets. The expression inside the brackets is a PERFECT SQUARE bcoz it is in the form of (root_a)^2*(x^2) + 2*a*k*(x) + k^2
so effectively in our example the perfect square in the bracket is

(2*x)^2 + 2 * 2 * (4.5) *x + (4.5)^2
further simplified as

(2*x + 4.5) ^2..........that is a perfect square!!!!!!!!!

(2*x)^2

Answer 4

well um basically look at this

X ^ 2 + (2) (x) (y) + Y ^ 2

x squared plus two times the square root of x ^ 2 times the square root of y ^2 ( or x and y) plus y squared.

And um well anytime you see that there is a square such as X in this example to complete the square what you got to do it look for a Y that when square will equal the middle term for example...

X ^ 2 + 4x Complete the square Aye

So you um like say maby this is really

X ^ 2 + 2 times (something) times (x)

Why not try 2 for something it's worth a shot.

X ^ 2 * 2 (2)x + 4
(4) or 2 squared or Y ^ 2 Something ^ 2

so clean this **** up and
(x + 2) ^ 2

Ahh much nicer and we see it is a parabola like when you throw something it falls in a parablola

yeah something like that
so .......

X ^ 2 + 6x + Something? complete the square by finding something now why don't you now?

then you find the limit of X as it approaches c of the function x which is x squared divided by x squared plus one.
yeah.... math......

Answer 5

(a+b) squared = a squared + 2ab + b squared
(a-b) squared = a squared - 2ab + b squared
a squared - b squared = (a+b)(a-b)
By using above three facts, we try to find the factors.

Completing the square technique is used for finding out the factors of an algebraic expression to which this technique applies.

For instance, x squared + 14x + 13, can be factored as:
= x squared + 14x + 49 -36 = (x+7) squared - (6) squared
= (x+7+6)(x+7-6) = (x+13)(x+1) Ans.

Answer 6

Completing the Square:
Solving Quadratic Equations
http://www.purplemath.com/modules/sqrquad.htm

Completing the Square
http://mathworld.wolfram.com/CompletingtheSquare.html

http://math.about.com/od/quadraticequation/Quadratic_Equation_Tutorials_Solvers.htm
http://math.about.com/library/q14.pdf

Answer 7

square

Answer 8

Why dont u refer a textbook?

Answer 9

weve been studying it you know its fun i think you should buy a text book for it algebra book

Answer 10

we all wont understand ur question coz its incomplete.