Solve by Factoring

Problem

√(,4*y+1)−√(,y−2)=3

Solution

1. Isolate one of the radical terms by adding √(,y−2) to both sides of the equation.

√(,4*y+1)=3+√(,y−2)

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

(√(,4*y+1))2=(3+√(,y−2))2

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

4*y+1=9+6√(,y−2)+(y−2)

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

4*y+1=y+7+6√(,y−2)

5. Isolate the remaining radical term by subtracting y and 7 from both sides.

3*y−6=6√(,y−2)

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

y−2=2√(,y−2)

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

(y−2)2=(2√(,y−2))2

8. Expand and simplify both sides of the equation.

y2−4*y+4=4*(y−2)

y2−4*y+4=4*y−8

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

y2−8*y+12=0

10. Factor the quadratic expression by finding two numbers that multiply to 12 and add to −8

(y−2)*(y−6)=0

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

y−2=0⇒y=2

y−6=0⇒y=6

12. Check for extraneous solutions by substituting both values back into the original equation. For y=2 √(,9)−√(,0)=3 which is 3=3 For y=6 √(,25)−√(,4)=3 which is 5−2=3 Both are valid.

Final Answer

√(,4*y+1)−√(,y−2)=3⇒y=2,6

Want more problems? Check here!