completing the square?

Question

the problem is 3x^2+14x=5
and i have to solve by completing the square.
help?
can you please tell me the steps because i actually want to know how to do these kinds of problems if i encounter them again. thanks a lot !

Answer 1

To complete a square, you must have a binomial square on one side of an equation... that is to say, it must be in the form of
x² + 2nx + n².

The key to solving using this method is the "middle term."

You're starting with 3x² + 14x = 5.
First, divide out that 3... the coefficient of the x² term.

x² + (14/3)x = 5/3. Rewrite this as:
x² + (14/3)x + ____ = 5/3 + ____

That "middle term" is twice the root you're looking for. (2nx)
Half of (14/3) is (7/3), so you're going to be shooting for
(x + 7/3)² on the left side of your equation. Fill in the blanks with (7/3)².

x² + (14/3)x + ____ = 5/3 + ____
x² + (14/3)x + (7/3)² = 5/3 + (7/3)²
x² + (14/3)x + (49/9) = 5/3 + (49/9)
x² + (14/3)x + (49/9) = 15/9 + 49/9 [Getting common denominators]
x² + (14/3)x + (49/9) = 64/9

Now you have a perfect square trinomial on the left. Factor it.
(x + 7/3)² = 64/9
Take the square root of both sides, then solve for x.
|x + 7/3| = 8/3
x + 7/3 = ±8/3
x = -7/3 ± 8/3
x = 1/3 or x = -15/3 = -5

Checking solutions,
3(1/9) + 14(1/3) = 1/3 + 14/3 = 5.
3(25) + 14(-5) = 75 - 70 = 5.

Both check, so x = 1/3 or x = -5 are both valid.

Completing a square is a longer method to solve quadratics than either factoring or using the quadratic formula, but you'll need to know how to complete a square when you turn to conic sections in a high school Algebra II course or in a College Algebra course.

A way to remember it easily is "half going down, square going up."
Two lines of your solution will look like this:

x² + ( n )x + ___
(x + ___)²

Whatever the "middle term" (or "n") is, half that number will fill in the blank below it, and the square of that will fill in the blank above.

Answer 2

First, you have to get rid of the coefficient in front of x^2, so divide everything by 3.

x^2 + (14/3)x = (5/3)

Then you divide the coefficient of x by 2 and square it, and the add that to both sides of the equation.

So you are taking 1/2 of 14/3 to get 7/3, and squaring it, to get 49/9.

x^2 +(14/3)x +(49/9) = (5/3) + (49/9)

Simplify
x^2 +(14/3)x +(49/9) = 64/9

Then you can factor the right side, it is now a perfect square trinomial.

(x + 7/3)^2 = 64/9

*Note that the number inside the parenthesis is the number you squared ealrier. (7/3)

Now take the square root of both sides. Both the positive and negative values.

(x + 7/3) = 8/3 or -(x + 7/3) = 8/3

Then solve for x.

x = 1/3 or -x - 7/3 = 8/3
-x = 15/3
-x = 5
x = -5
So the steps without the explanation look like this.

x^2 + (14/3)x = (5/3)

x^2 +(14/3)x +(49/9) = (5/3) + (49/9)

x^2 +(14/3)x +(49/9) = 64/9

(x + 7/3)^2 = 64/9

(x + 7/3) = 8/3 or -(x + 7/3) = 8/3

x = 1/3 or -x - 7/3 = 8/3
................. -x = 15/3
................. -x = 5
...................x = -5

Your final answer is 1/3 or 5
Good Luck

Answer 3

1) Divide out a "3":

3(x^2 + (14/3)x) = 5

2) Take the "(14/3)x" term and divide the coefficient by 2

(14/3) / 2 = 7/3

3) Find the square of 7/3
(7/3)^2 = 49/9

4) You've found the number to add to (x^2 + (14/3)x) to get a perfect square trinomial, so add it to the terms:

3(x^2 + (14/3)x + (49/9))

5) Since you've added it to one side of the equation, add it to the other. But, since it is in the parentheses, multiply it by the "3" outside of the parentheses first:

3(x^2 + (14/3)x + (49/9)) = 5 + 3*(49/9)

6) Write the trinomial as the square of a binomial:

3 (x + (7/3))^2 = 5 + (49/3)
3 (x + (7/3))^2 = (64/3)
(x + (7/3))^2 = (64/9)


7)Take square roots of each side and solve(remember if you take a square root, the answer is either negative or positive):

x + (7/3) = +/-(8/3)
x = -5 or 1/3

Answer 4

"Completing the square" means you want to manipulate the equation to get (x+something)^2=nicesquarenumber. The first step is to get the x-squared coefficient to equal 1, which in this case means dividing by 3.


x^2 + (14/3)x = (5/3)

Then, take half of the x term and square it. Add this amount to both sides.

x^2 + (14/3)x + (49/9) = (5/3) + (49/9)
x^2 + (14/3)x + (49/9) = (64/9)

Now, the left hand side of this equation can be converted into squared form. To figure out what goes in the parentheses with the x, remember that it is the square root of that term you just added.

(x + 7/3)^2 = 64/9

Well, we can easily take the square root of both of those sides and solve for x.

x + 7/3 = 8/3
x = 8/3 - 7/3
x = 1/3

Hope that helps!

Answer 5

THIS IS THE METHOD ON A SIMPLIER EXAMPLE, STEP BY STEP. Complete the square can be done is 3 steps.

Ok, this example may be a little too complex to write here. I will start with a simpler example, and go on to solve your question using principles gleaned from simple example.

Consider, 2x^2 + 6x = 1, you want to complete the square.
1. Make sure that coefficient of (x^2) is 1 -- to get this, I divide the equation above throughout by 2: x^2 + 3x = 1/2

2. Add and subtract then SQUARE of HALF the coefficient of (x) (Added tems in SQUARE BRACKETS [])
So from x^2 + 3x = 1/2, I get x^2 + 3x + [(3/2)^2 - (3/2)^2] = 1/2
As you can see from the above, I added exactly ZERO to the equation

3. You're ready to complete the square! Remember how (x+a)^2 = x^2 + 2ax + a^2? In the case above, a is (3/2)
Thus from above, you get (x + 3/2)^2 - (3/2)^2 = 1/2
Simplying further, you get (x + 3/2)^2 = 1/2 + (3/2)^2
(x+3/2)^2 = 11/4

OK, now that we have the basics, let's apply them to the question

3x^2 + 14x = 5
1. ensure that the coef of (x^2) is 1, so divide throughout by 3
x^2 + (14/3)x = 5/3

2. Add and subtract the square of half the coef of x,
x^2 + (14/3)x + (14/6)^2 - (14/6)^2 = 5/3

3. Compelte the sqaure!
(x + 14/6)^2 - (14/6)^2 = 5/3
(x + 14/6)^2 = 64/9

Answer 6

NOw consider the equation 3x^2 + 14x = 5
divide the equation by 3, we get x^2 + 14/3(x) = 5/3
Now add half of 14/3 squared to both sides ,(14/3 is the coefficient of x), we get
x^2 + 14/3(x) + (7/3)^2 = 5/3 + (7/3)^2

or (x + 7/3)^2 = 64/9

or (x + 7/3) = + or - 8/3

or x = 8/3 - 7/3 or -8/3 - 7/3

= 1/3 or -5

Answer 7

To use "completing the square" method it's easiest when you have the form (ax)^2 +bx+c=d. Since you have a coefficient 3 in front of x^2 term, it might be easier to divide both sides by 3, that gives:
x^2+(14/3)x=5/3... Remember the formula: a^2+2ab+b^2=(a+b)^2...then your a is x and your b is 7/3 so:
x^2+2(x)(7/3)+(7/3)^2=5/3+(7/3)^2
(x+7/3)^2=64/9
so x+7/3=8/3 or x+7/3=-8/3
x=1/3 or x=-5

Answer 8

3x^2 + 14x = 5
divide everything by 3
x^2 + (14/3)x = (5/3)
find half of (14/3), square it, add to both sides
x^2 + (14/3)x + (49/9) = (64/9)
factor the left into a perfect square
(x + (7/3))^2 = (64/9)
sqrt both sides
x + (7/3) = ±(8/3)
x = (-7/3) ± (8/3)
x = (-15/3) or (1/3)
x = -5 or (1/3)

Answer 9

1) take 1/2 of the coefficient of x.
2) square that
3) Add the result to both sides of the equation.
4) Square root the equation
5) Solve

Answer 10

3x^2 + 14x = 5, 3x^2 + 14x - 5 = 0

If ax^2 + bx + c = 0 then x1 = (-b + sqrt(b^2 - 4ac))/2a and
x2 = (-b - sqrt(b^2 -4ac))/2a (where sqrt means square root)

so

x1 = (-14 + sqrt(14^2 -4x3x(-5))/2x3 = (-14 + sqrt(196 + 60))/6 =
(-14 + sqrt(256))/6 = (-14 + 16)/6 = 2/6 = 1/3, and

x2 = (-14 - sqrt(14^2 - 4x3x(-5))/2x3 = . . . = (-14 -16)/6 = -30/6 = -5