Question

6x2 + 19x + 10 = 0 find roots of the quadratic equation by the method of completing square​

Answer 1

Answer:

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Step-by-step explanation:

Divide both sides of the equation by 6 to have 1 as the coefficient of the first term :

x2+(19/6)x+(5/3) = 0

Subtract 5/3 from both side of the equation :

x2+(19/6)x = -5/3

Now the clever bit: Take the coefficient of x , which is 19/6 , divide by two, giving 19/12 , and finally square it giving 361/144

Add 361/144 to both sides of the equation :

On the right hand side we have :

-5/3 + 361/144 The common denominator of the two fractions is 144 Adding (-240/144)+(361/144) gives 121/144

So adding to both sides we finally get :

x2+(19/6)x+(361/144) = 121/144

Adding 361/144 has completed the left hand side into a perfect square :

x2+(19/6)x+(361/144) =

(x+(19/12)) • (x+(19/12)) =

(x+(19/12))2

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

x2+(19/6)x+(361/144) = 121/144 and

x2+(19/6)x+(361/144) = (x+(19/12))2

then, according to the law of transitivity,

(x+(19/12))2 = 121/144

We'll refer to this Equation as Eq. #5.2.1

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

Note that the square root of

(x+(19/12))2 is

(x+(19/12))2/2 =

(x+(19/12))1 =

x+(19/12)

Now, applying the Square Root Principle to Eq. #5.2.1 we get:

x+(19/12) = √ 121/144

Subtract 19/12 from both sides to obtain:

x = -19/12 + √ 121/144

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

x2 + (19/6)x + (5/3) = 0

has two solutions:

x = -19/12 + √ 121/144

or

x = -19/12 - √ 121/144

Note that √ 121/144 can be written as

√ 121 / √ 144 which is 11 / 12