Check whether the equation 5x² – 6x – 2 = 0 has real roots and if it has, find them by the method of completing the square. Also verify that roots obtained satisfy the given equation.

We have the equation, 5x² – 6x – 2 = 0

Comparing this equation by ax² + bx + c = 0, we get

a = 5

b = -6

& c = -2

To test whether or not the equation has real roots, put these values in discriminant D. We get

D = b² – 4ac

⇒ D = (-6)² – 4(5)(-2)

⇒ D = 36 + 40 = 76

Since D>0, this means the equation will have 2 real roots.

Recalling equation,

5x² – 6x – 2 = 0

Now finding the roots by completing the square,

Dividing the equation by 5 throughout, we get

⇒ …(i)

We know,

(x – y)² = x² – 2xy + y² …(ii)

Comparing equations (i) & (ii), we can write as

⇒

⇒

Adding and subtracting from equation (i), we get

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒ or

⇒ or

So the roots has come out to be and .

The next step is to verify whether or not these roots satisfy the equation, 5x² – 6x – 2 = 0.

First, put in the equation.

⇒ [∵ (√19 + 3)² = 19 + 9 + 6√19]

⇒

⇒

⇒ 28 – 18 – 10 = 0 × 5

⇒ 10 – 10 = 0

⇒ 0 = 0

Since it is true, thus is a root of the equation 5x² – 6x – 2 = 0.

Now, put in the equation. We get,

⇒ [∵ (-√19 + 3)² = 19 + 9 – 6√19]

⇒

⇒

⇒ 28 – 18 – 10 = 0

⇒ 10 – 10 = 0

⇒ 0 = 0

Since it is true, thus is a root of the equation 5x² – 6x – 2 = 0.

Hence, verified.