Write x² - 2x - 3 = 0 in the form (x - a)² = b, where a and b are integers

Solution:

Given equation is x² - 2x - 3 = 0

Given quadratic equation can be written in the form (x - a)² = b by following steps:

Take the coefficient of the middle term (-2) and divide it by 2

Then square the result.

Add this number into the equation right after the middle term

Subtract this number after the last term.

Take the middle term and divide by 2……………..-2/2 = -1

Then square the result ……. (-1)² = 1

Add this number into the equation right after the middle term ……

let us proceed accordingly,

x² - 2x + 1 - 3 - 1 = 0

⇒ x² - 2x + 1 = 4

LHS term is in the form of x² - 2 × x × 1 + 1²

= a² - 2ab  +b²

But by using standard formula:

a² - 2ab + b² = (a - b)²

⇒x² - 2 × x × 1 + 1² = (x - 1)²

Therefore

x² - 2x + 1 = 4

⇒ (x - 1)² = 4.

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Write x² - 2x - 3 = 0 in the form (x - a)² = b, where a and b are integers

Summary:

The equation x² - 2x - 3 = 0 in the form (x - a)² = b can be written as (x - 1)² = 4.