how do you solve a quadratic by completing the square?

Question

please can someone take me step by step to a solution to the following equation using the "Completing the square" method:

8x^2 - 5x - 3 = 0

Answer 1

First you have to divide both sides by the leading coefficient (8, in this case).

x^2 - (5/8)x - 3/8 = 0

Now you add (1/2 * 2nd coefficient)^2 to both sides. The second coefficient is 5/8 so add (1/2 * 5/8)^2 to both sides.

x^2 - (5/8)x + (1/2 * 5/8)^2 - 3/8 = (1/2 * 5/8)^2
x^2 - (5/8)x + (25/256) - 3/8 = 25/256

Add 3/8 to both sides:
x^2 - (5/8)x + (25/256) = (25/256) + (3/8)

Factor the perfect square on the left:
(x - 5/16)^2 = (25/256) + (3/8)

Make a common denominator on the right:
(x - 5/16)^2 = (25/256) + (96/256)
(x - 5/16)^2 = 121/256

Take the square root of both sides:
x - 5/16 = ±sqrt(121/256)
x - 5/16 = ±11/16

Add 5/16 to both sides:
x = 5/16 ± 11/16

So
x = (5 ± 11) / 16

x = (5 + 11)/16 = 16/16 = 1
and
x = (5 - 11) / 16 = -6/16 = -3/8

x = -3/8 and 1

Answer 2

Divide through by 8

x^2 - (5/8)X - (3/8) = 0

Move the constant to the other side

x^2 - (5/8)X = (3/8)

Take 1/2 of the coefficient of X and square it.

(-5/16)^2 = (25/256)

Add this to both sides of the equation:

x^2 - (5/8)X + (25/256) = (3/8) + (25/256)

The left side is now a perfect square that can be expressed as:

(x - 5/16)^2 = (3/8) + (25/256)

Find a commeon denominator

(x - 5/16)^2 = (96/256) + (25/256)

(X - 5/16)^2 = 121/256

Take the square root of both sides:

(X - 5/16) = +/- 11/16

So, the answers are:

X = 5/16 + 11/16 = 16/16 = 1

and

X = 5/16 - 11/16 = -6/16 = - 3/8

Answer 3

8 [ x ² - (5/8) x - 3/8 ] = 0
[ x ² - (5/8) x + 25 / 256) ] - 25 / 256 - 3 / 8 = 0
(x - 5 / 16) ² = 25 / 256 + 3 / 8
(x - 5 / 16) ² = 25/256 + 96 / 256
(x - 5 / 16) ² = 121 / 256
x - 5 / 16 = ± 11 / 16
x = 5 / 16 ± 11 / 16
x = 16 / 16 , x = - 6 / 16
x = 1 , x = - 3 / 8

Check by factorising:-
(8x + 3) (x - 1) = 0
x = - 3 / 8 , x = 1 (as above)

Answer 4

8x^2 - 5x - 3 = 0
----------------------------

Keep in mind the identities:
(x + e)^2 = x^2 + 2ex + e^2
(x - e)^2 = x^2 - 2ex + e^2
----------------------------
If you have x^2 + bx, you need to add (b/2)^2
to complete the square.
x^2 + bx + (b/2)^2 = (x + b/2)^2
-----------------------------------------------
You proposed 8x^2 - 5x - 3 = 0

x^2 - 5x/8 - 3/8 = 0
x^2 - 5x/8 = 3/8
In order to generate a perfect square you need to add (5/16)^2
to both sides of the equation

x^2 - 5x/8 + (5/16)^2 = 3/8 + 25/256

(x - 5/4)^2 = 6/16 + 25/256 = 96/16 + 25/256 = 121/256 = (11/16)^2

Taking the square-roots lead to

x - 5/4 = 11/16 or x - 5/4 = -11/16

which give the solutions to the equation

x = 5/4 + 11/16 or x = 5/4 - 11/16

x = 31/16 or x = 9/16

Answer 5

Ok, let's give this problem a try,

- divide the equation by 8

x^2 - 5/8x - 3/8 = 0

- Analyze the x coefficient. The idea is to divide the coefficient by 2. You will get 5/16.

- Multiply 5/16 by 5/16 = 25/(16^2)

- Factor is now
(x-5/16)(x-5/16) - 3 - 25/(16^2)= 0

or (x-5/16)^2 = 3+ 25/(16^2)

Hope this explanation helps.

Answer 6

8x^2 - 5x - 3 = 0
x^2 -5/8x -3/8 = 0
x^2 -5x/8 +25/256 -3/8 = 25/256
(x-5/16)^2 = 25/256+3/8 = 121/256
x-5/16 = +/- sqrt(121/256) = +/- 11/16
x = 5/16 +/- 11/16
x = 1 or -3/8