Question

Suppose a and b are the solutions to the quadratic equation 2x^2-3x-6=0. Find the value of (a+2)(b+2).

Answer 1

(a+2)(b+2) = 4

Explanation:

We are given the following quadratic equation:

`2x^2-3x-6=0`

Let a a and b be the solution of the given quadratic equation.

Solving the equation:

`2x^2-3x-6=0\\\text{Using the quadratic formula}\\\\x = (-b \pm √(b^2-4ac))/(2a)\\\\\text{Comparing the equation to }ax^2 + bx + c = 0\\\text{We have}\\a = 2\\b = -3\\c = -6\\x = (3\pm √(9-4(2)(-6)))/(4) = (3\pm √(57))/(4)\\\\a = (3+√(57))/(4), b = (3-√(57))/(4)`

We have to find the value of (a+2)(b+2).

Putting the values:

`(a+2)(b+2)\\\\=\bigg((3+√(57))/(4)+2\bigg)\bigg((3-√(57))/(4)+2\bigg)\\\\=\bigg((11+√(57))/(4)\bigg)\bigg((11-√(57))/(4)\bigg)\\\\=(121-57)/(156) = (64)/(16) = 4`