Simplify ( square root of x^2+9-5)/(x+4)

Problem

(√(,x2+9)−5)/(x+4)

Solution

1. Identify the expression as a fraction involving a radical in the numerator.

2. Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is √(,x2+9)+5

(√(,x2+9)−5)/(x+4)⋅(√(,x2+9)+5)/(√(,x2+9)+5)

3. Expand the numerator using the difference of squares formula (a−b)*(a+b)=a2−b2

((√(,x2+9))2−5)/((x+4)*(√(,x2+9)+5))

4. Simplify the terms in the numerator.

(x2+9−25)/((x+4)*(√(,x2+9)+5))

5. Combine the constants in the numerator.

(x2−16)/((x+4)*(√(,x2+9)+5))

6. Factor the numerator using the difference of squares x2−16=(x−4)*(x+4)

((x−4)*(x+4))/((x+4)*(√(,x2+9)+5))

7. Cancel the common factor (x+4) from the numerator and the denominator, assuming x≠−4

(x−4)/(√(,x2+9)+5)

Final Answer

(√(,x2+9)−5)/(x+4)=(x−4)/(√(,x2+9)+5)

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