Solve by Factoring

Problem

√(,4*y+20)−√(,y−4)=6

Solution

1. Isolate one of the radical terms by adding √(,y−4) to both sides of the equation.

√(,4*y+20)=6+√(,y−4)

2. Square both sides of the equation to begin removing the radicals.

(√(,4*y+20))2=(6+√(,y−4))2

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

4*y+20=36+12√(,y−4)+(y−4)

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

4*y+20=32+y+12√(,y−4)

5. Isolate the remaining radical term by subtracting y and 32 from both sides.

3*y−12=12√(,y−4)

6. Divide both sides by the common factor 3 to simplify the coefficients.

y−4=4√(,y−4)

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

(y−4)2=(4√(,y−4))2

8. Expand and simplify both sides of the equation.

y2−8*y+16=16*(y−4)

y2−8*y+16=16*y−64

9. Rearrange the equation into standard quadratic form a*y2+b*y+c=0 by moving all terms to one side.

y2−24*y+80=0

10. Factor the quadratic expression by finding two numbers that multiply to 80 and add to −24

(y−20)*(y−4)=0

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

y=20

y=4

12. Check for extraneous solutions by substituting the values back into the original equation. For y=20 √(,100)−√(,16)=10−4=6 (True). For y=4 √(,36)−√(,0)=6−0=6 (True).

Final Answer

√(,4*y+20)−√(,y−4)=6⇒y=4,20

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