If a and b are the roots of equation 2x2 + x + 2 = 0, then the value of  ab2+ba2𝑎𝑏2+𝑏𝑎2 is

The roots of the equation 2x² + x + 2 = 0 can be found using the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a.

Here, a = 2, b = 1, and c = 2.

Substituting these values into the quadratic formula gives:

x = [-1 ± sqrt((1)² - 422)] / 2*2 x = [-1 ± sqrt(1 - 16)] / 4 x = [-1 ± sqrt(-15)] / 4

Since the term under the square root is negative, the roots of the equation are complex numbers.

Let's denote the roots as a and b. So, a = (-1 + sqrt(-15)) / 4 and b = (-1 - sqrt(-15)) / 4.

Now, we need to find the value of ab² + ba².

Substituting the values of a and b into this expression gives:

ab² + ba² = [(-1 + sqrt(-15)) / 4][(-1 - sqrt(-15)) / 4]² + [(-1 - sqrt(-15)) / 4][(-1 + sqrt(-15)) / 4]²

Solving this expression will give the required value. However, since the roots are complex numbers, the calculation will be quite complex.

But, we can use the Vieta's formulas which states that for a quadratic equation ax² + bx + c = 0, the product of the roots is c/a.

So, ab = c/a = 2/2 = 1.

Substituting ab = 1 into the expression ab² + ba² gives:

ab² + ba² = 1b² + 1a² = a² + b²

So, the value of ab² + ba² is equal to the sum of the squares of the roots of the equation.

To find the exact value, we would need to calculate the squares of the complex roots, which would involve complex arithmetic.