Solve by Factoring

Problem

√(,4*y+12)−√(,y−6)=6

Solution

1. Isolate one of the radical terms to make squaring easier.

√(,4*y+12)=6+√(,y−6)

2. Square both sides of the equation to eliminate the first radical.

(√(,4*y+12))2=(6+√(,y−6))2

3. Expand the right side using the identity (a+b)2=a2+2*a*b+b2

4*y+12=36+12√(,y−6)+(y−6)

4. Simplify the equation by combining like terms on the right side.

4*y+12=y+30+12√(,y−6)

5. Isolate the remaining radical term by moving all other terms to the left side.

3*y−18=12√(,y−6)

6. Divide both sides by 3 to simplify the coefficients before squaring again.

y−6=4√(,y−6)

7. Square both sides again to eliminate the final radical.

(y−6)2=(4√(,y−6))2

8. Expand and simplify the resulting equation.

y2−12*y+36=16*(y−6)

y2−12*y+36=16*y−96

9. Rearrange the equation into standard quadratic form a*y2+b*y+c=0

y2−28*y+132=0

10. Factor the quadratic equation by finding two numbers that multiply to 132 and add to −28

(y−6)*(y−22)=0

11. Solve for y by setting each factor to zero.

y=6

y=22

12. Check for extraneous solutions by substituting both values into the original equation. For y=6 √(,36)−√(,0)=6 which is 6=6 (True). For y=22 √(,100)−√(,16)=10−4=6 (True).

Final Answer

√(,4*y+12)−√(,y−6)=6⇒y=6,22

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