Solve by Factoring square root of 9x+67=x+5

Problem

√(,9*x+67)=x+5

Solution

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

(√(,9*x+67))2=(x+5)2

9*x+67=x2+10*x+25

2. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting 9*x and 67 from both sides.

0=x2+10*x−9*x+25−67

x2+x−42=0

3. Factor the quadratic expression by finding two numbers that multiply to −42 and add to 1

(x+7)*(x−6)=0

4. Apply the zero product property to find the potential solutions for x

x+7=0⇒x=−7

x−6=0⇒x=6

5. Check for extraneous solutions by substituting the values back into the original radical equation.

For *x=−7:√(,9*(−7)+67)=√(,4)=2; Right side: −7+5=−2.* (Extraneous)

For *x=6:√(,9*(6)+67)=√(,121)=11; Right side: *6+5=11.* (Valid)

Final Answer

√(,9*x+67)=x+5⇒x=6

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