Solve for x square root of x+6=x-6

Problem

√(,x+6)=x−6

Solution

1. Square both sides of the equation to eliminate the radical on the left side.

√(,x+6)2=(x−6)2

x+6=(x−6)2

2. Expand the right side using the binomial square formula (a−b)2=a2−2*a*b+b2

x+6=x2−12*x+36

3. Rearrange into a quadratic equation by moving all terms to one side to set the equation to zero.

0=x2−12*x−x+36−6

x2−13*x+30=0

4. Factor the quadratic expression by finding two numbers that multiply to 30 and add to −13

(x−10)*(x−3)=0

5. Solve for x by setting each factor equal to zero.

x−10=0⇒x=10

x−3=0⇒x=3

6. Check for extraneous solutions by substituting each value back into the original equation.

For *x=10:√(,10+6)=10−6⇒√(,16)=4⇒4=4* (True)

For *x=3:√(,3+6)=3−6⇒√(,9)=−3⇒3=−3* (False)

Final Answer

√(,x+6)=x−6⇒x=10

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