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Bot GPT 3.5 To find the domain of √(2x^2 + 3x - 12), we need to determine the values of x for which the expression under the square root is non-negative (i.e., greater than or equal to zero) since we cannot take the square root of a negative number.
The expression under the square root, 2x^2 + 3x - 12, needs to be greater than or equal to zero:
2x^2 + 3x - 12 ≥ 0
Factoring the quadratic equation gives:
(2x - 3)(x + 4) ≥ 0
The roots of the equation are x = 3/2 and x = -4.
To determine the sign of the expression (2x - 3)(x + 4) in the intervals created by the roots, choose a test point in each interval:
- If x < -4, pick x = -5:
(2(-5) - 3)(-5 + 4) = (-13)(-1) = 13 > 0
- If -4 < x < 3/2, pick x = 0:
(2(0) - 3)(0 + 4) = (-3)(4) = -12 < 0
- If x > 3/2, pick x = 2:
(2(2) - 3)(2 + 4) = (1)(6) = 6 > 0
From this assessment, we can see that the expression (2x - 3)(x + 4) is negative for -4 < x < 3/2, and it is non-negative for x ≤ -4 and x ≥ 3/2.
Therefore, the domain of √(2x^2 + 3x - 12) is x ≤ -4 or x ≥ 3/2.