Solve the equation by completing the square

Question

x² -4 x – 8 = 0

Answer 1

Our equation is x² - 4x - 8 = 0

"Completing the square" means trying to write the x² - 4x - 8 as the product of two identical factors:

x² - 4x - 8 = (x + something)(x + something).

We have to find out what this "something" is (and then we might have to change the x² - 4x - 8 a little bit, in order to accommodate the "something" that we find).

Let's multiply out the (x + something)(x + something) and see what we get.

(x + something)(x + something) = x² + 2x·something + something²

[Are you okay with the above multiplication? The multiplication yields four terms,

(x + something)(x + something) = x² + x·something + x·something + something²

and I combined the two center terms (x·something + x·something) into 2x·something.]

We want the above product x² + 2x·something + something² to be equal to x² - 4x - 8.

Therefore the 2x·something term must be equal to the -4x term:

2x·something = -4x

To find out what the something is, divide both sides of the above equation by 2x:

something = (-4x)/(2x) = -2.

So the "something" is -2.

Therefore our two identical factors (x + something)(x + something) are really (x-2)(x-2).

Let's multiply out the (x-2)(x-2) and see what we get:

(x-2)(x-2) = x² - 4x + 4.

This x² - 4x + 4 is almost the same as the polynomial x² - 4x - 8 that we started with.

To get the x² - 4x + 4, all we have to do is add 12 to both sides of the x² - 4x - 8 = 0

x² - 4x - 8 = 0

+12 +12

-----------------------

x² - 4x + 4 = 12

Now we have a left side that we can easily factor into two identical factors (and we already know what the two factors are):

(x-2)(x-2) = 12

The "x" appears in two different places in the above equation. To make the "x" appear in only one place,

take the square root of both sides. (Since the left side is the product of two identical factors, it's very easy to take the square root of the left side.)

x-2 = ±√12

To get the "x" all by itself, add 2 to both sides of the above equation:

x = 2 ± √12

If desired, you can simplify the √12 like this.

√12 = √(4·3) = √(2²·3) = 2√3

("Simplifying" means getting the smallest possible number under the square root sign. For example, 2√3 is simpler than √12 because 3 is smaller than 12.)

So the answer x = 2 ± √12 can also be written as 2 ± 2√3

Answer 2

x^2 -4x - 8 =0

x^2 -4x +4 = 8+4 =12

(x-2)^2= 12

x-2 = +/-sqr12

x = 2 +/-2sqr3

x=2+2sqr3 or 2-2sqr3

x = about 5.46 or - 1.46

Answer 3

Step 1: Get rid of the "c" term, which is -8 by adding it on both sides. x² -4x = 8

Step 2: Add a number that will make the left-hand-side a perfect square trinomial. Use the formula (b/2)². This gives us x² -4x + 4 = 8+4. (This is how we get 4: we know b=-4, so b/2 is -2, and (-2)² =4.)

Step 3: Represent the left-hand-side in factored form, which is (x-2)(x-2).

Step 4: We represent the left-hand-side as a square so that we can take the square root of both sides later. (x-2)² = 12.

Step 5: We take the square root of both sides. √(x-2)² = ±√12

Step 6: We simplify both sides. The square root and the square cancel each other on the left-hand-side. The square root on the right-hand-side simplifies to ±√3√4 = ±2√3. This gives us x-2 = ±2√3.

Step 7: Get x alone by adding 2 to both sides: x = 2 ± 2√3.