Question

what is the answer of x²-5x+19​

Answer 1

Solving -x2-5x-19 = 0 by Completing The Square .

Multiply both sides of the equation by (-1) to obtain positive coefficient for the first term:

x2+5x+19 = 0 Subtract 19 from both side of the equation :

x2+5x = -19

Now the clever bit: Take the coefficient of x , which is 5 , divide by two, giving 5/2 , and finally square it giving 25/4

Add 25/4 to both sides of the equation :

On the right hand side we have :

-19 + 25/4 or, (-19/1)+(25/4)

The common denominator of the two fractions is 4 Adding (-76/4)+(25/4) gives -51/4

So adding to both sides we finally get :

x2+5x+(25/4) = -51/4

Adding 25/4 has completed the left hand side into a perfect square :

x2+5x+(25/4) =

(x+(5/2)) • (x+(5/2)) =

(x+(5/2))2

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

x2+5x+(25/4) = -51/4 and

x2+5x+(25/4) = (x+(5/2))2

then, according to the law of transitivity,

(x+(5/2))2 = -51/4

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

Note that the square root of

(x+(5/2))2 is

(x+(5/2))2/2 =

(x+(5/2))1 =

x+(5/2)

Now, applying the Square Root Principle we get:

x+(5/2) = √ -51/4

Subtract 5/2 from both sides to obtain:

x = -5/2 + √ -51/4

In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1

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

x2 + 5x + 19 = 0

has two solutions:

x = -5/2 + √ 51/4 • i

or

x = -5/2 - √ 51/4 • i

Note that √ 51/4 can be written as

√ 51 / √ 4 which is √ 51 / 2