Solve by Factoring

Problem

√(,4*y+5)−√(,y−1)=3

Solution

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

√(,4*y+5)=3+√(,y−1)

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

(√(,4*y+5))2=(3+√(,y−1))2

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

4*y+5=9+6√(,y−1)+(y−1)

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

4*y+5=y+8+6√(,y−1)

5. Isolate the remaining radical term by subtracting y and 8 from both sides.

3*y−3=6√(,y−1)

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

y−1=2√(,y−1)

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

(y−1)2=(2√(,y−1))2

8. Expand and simplify both sides of the equation.

y2−2*y+1=4*(y−1)

y2−2*y+1=4*y−4

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

y2−6*y+5=0

10. Factor the quadratic expression by finding two numbers that multiply to 5 and add to −6

(y−5)*(y−1)=0

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

y−5=0⇒y=5

y−1=0⇒y=1

12. Verify the solutions in the original equation to check for extraneous roots. For y=5 √(,25)−√(,4)=5−2=3 For y=1 √(,9)−√(,0)=3−0=3 Both are valid.

Final Answer

√(,4*y+5)−√(,y−1)=3⇒y=1,5

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