Rewrite the equation by completing the square.

2x^2 - 3x-5=0

2[ (x - 3/4) ^2 - 49/16]

Step-by-step explanation:

Factor 2x^2 - 3x-5=0 as follows:

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

Now focus on x^2 - (3/2) x - 5/2 alone.

To complete the square, take half of the coeficient of x: (1/2) (-3/2, or

-3/4. Now square this, obtaining 9/16.

Going back to x^2 - (3/2) x - 5/2, add in 9/16 and then subtract 9/16:

x^2 - (3/2) x + 9/16 - 9/16 - 5/2

Rewrite x^2 - (3/2) x + 9/16 as the square of a binomial: (x - 3/4) ^2

Then we have: (x - 3/4) ^2 - 9/16 - 5/2, or

(x - 3/4) ^2 - 9/16 - 40/16, or

(x - 3/4) ^2 - 49/16

Now go back to the original equation, 2x^2 - 3x-5=0, recall that this is equivalent to 2 (x^2 - (3/2) - 5/2). Replace x^2 - (3/2) - 5/2) with

(x - 3/4) ^2 - 49/16, obtaining 2[ (x - 3/4) ^2 - 49/16]