Question

Solve the eq. n square -3n =9
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Answer 1

$\huge\purple{Required \: Answer :}$

To Solve :

${n}^{2} - 3n = 9$

Solution :

${n}^{2} - 3n = 9 \\ {n}^{2} - 3n - 9 = 0$

Solving n2-3n-9 = 0 by Completing The Square .

Add 9 to both side of the equation :

n2-3n = 9

Now the clever bit: Take the coefficient of n , which is 3 , divide by two, giving 3/2 , and finally square it giving 9/4

Add 9/4 to both sides of the equation :

On the right hand side we have :

9 + 9/4 or, (9/1)+(9/4)

The common denominator of the two fractions is 4 Adding (36/4)+(9/4) gives 45/4

So adding to both sides we finally get :

n2-3n+(9/4) = 45/4

Adding 9/4 has completed the left hand side into a perfect square :

n2-3n+(9/4) =

(n-(3/2)) • (n-(3/2)) =

(n-(3/2))2

Things which are equal to the same thing are also equal to one another. Since

n2-3n+(9/4) = 45/4 and

n2-3n+(9/4) = (n-(3/2))2

then, according to the law of transitivity,

(n-(3/2))2 = 45/4

We'll refer to this Equation as Eq. #2.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

(n-(3/2))2 is

(n-(3/2))2/2 =

(n-(3/2))1 =

n-(3/2)

Now, applying the Square Root Principle to Eq. #2.2.1 we get:

n-(3/2) = √ 45/4

Add 3/2 to both sides to obtain:

n = 3/2 + √ 45/4

Since a square root has two values, one positive and the other negative

n2 - 3n - 9 = 0

has two solutions:

n = 3/2 + √ 45/4

or

n = 3/2 - √ 45/4

Note that √ 45/4 can be written as

√ 45 / √ 4 which is √ 45 / 2