Solve for x square root of x+14=x-6

Problem

√(,x+14)=x−6

Solution

1. Square both sides of the equation to eliminate the radical.

√(,x+14)2=(x−6)2

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

x+14=x2−12*x+36

3. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting x and 14 from both sides.

0=x2−13*x+22

4. Factor the quadratic expression by finding two numbers that multiply to 22 and add to −13

0=(x−11)*(x−2)

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

x=11

x=2

6. Check for extraneous solutions by substituting the values back into the original equation. For x=11 √(,11+14)=11−6⇒√(,25)=5 which is true. For x=2 √(,2+14)=2−6⇒√(,16)=−4 which is false because the principal square root is positive.

Final Answer

x=11

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