What number should be added to both sides of the equation to complete the square? x² - 6x - 13 = 0

Solution:

We will express the given equation in the form (x - a)² = 0.

Let us see what could be added to both sides of this equation to complete the square.

On expanding (x - a)² = 0, we get x² - 2ax + a² = 0

On comparing x² - 2ax + a² = 0 with x² - 6x - 13 = 0, we get -2a = -6.

This means the value of a is 3.

Now, we have to decide a number that should be added to both sides of equation x² - 6x - 13 = 0 to make LHS equivalent to (x - 3)².

x² - 6x - 13 = 0

(x)² - 2 × (x) × 3 = 13

If we add (3)² on both sides, we get,

(x)² - 2 × (x) × 3 + (3)² = 13 +  (3)² 

x² - 6x + 9 = 13 + 9

(x - 3)² = 22

Therefore, we should add 22 to both sides of the equation x² - 6x - 13 = 0 to complete the square.

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What number should be added to both sides of the equation to complete the square? x² - 6x - 13 = 0

Summary:

The number 22 should be added to both sides of the equation x² - 6x - 13 = 0 to complete the square.