Find the Domain x+6 = square root of 18+3x+x^2

Problem

x+6=√(,18+3*x+x2)

Solution

1. Identify the restriction for a square root function. For the expression to be defined in the set of real numbers, the radicand (the expression inside the square root) must be greater than or equal to zero.

18+3*x+x2≥0

2. Rearrange the quadratic expression into standard form.

x2+3*x+18≥0

3. Determine the discriminant of the quadratic a*x2+b*x+c using the formula D=b2−4*a*c to find the roots.

D=3−4*(1)*(18)

D=9−72

D=−63

4. Analyze the result. Since the discriminant is negative (D<0 and the leading coefficient is positive (1>0, the parabola y=x2+3*x+18 opens upward and never touches or crosses the xaxis.

5. Conclude that the quadratic expression x2+3*x+18 is always positive for all real values of x Therefore, there are no restrictions imposed by the square root.

6. Consider the domain of the equation as a whole. In the context of solving the equation, the left side x+6 must also be non-negative because it equals a principal square root, but the "domain" of the expression/function itself refers to the set of all possible input values for which the expressions are defined. Since both sides are defined for all real numbers, the domain is all real numbers.

Final Answer

Domain: *(−∞,∞)

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