Solve by Factoring

Problem

√(,4*y+16)−√(,y−5)=6

Solution

1. Isolate the first radical by adding √(,y−5) to both sides of the equation.

√(,4*y+16)=6+√(,y−5)

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

(√(,4*y+16))2=(6+√(,y−5))2

4*y+16=36+12√(,y−5)+(y−5)

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

4*y+16=31+y+12√(,y−5)

4. Isolate the remaining radical term on one side.

3*y−15=12√(,y−5)

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

y−5=4√(,y−5)

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

(y−5)2=(4√(,y−5))2

y2−10*y+25=16*(y−5)

7. Expand and move all terms to one side to set the quadratic equation to zero.

y2−10*y+25=16*y−80

y2−26*y+105=0

8. Factor the quadratic equation by finding two numbers that multiply to 105 and add to −26

(y−5)*(y−21)=0

9. Solve for y by setting each factor equal to zero.

y−5=0⇒y=5

y−21=0⇒y=21

10. Check for extraneous solutions by substituting both values back into the original equation.

For *y=5:√(,4*(5)+16)−√(,5−5)=√(,36)−0=6

For *y=21:√(,4*(21)+16)−√(,21−5)=√(,100)−√(,16)=10−4=6

Final Answer

√(,4*y+16)−√(,y−5)=6⇒y=5,21

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