Solve by Factoring

Problem

√(,2*x+3)−√(,x+1)=1

Solution

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

√(,2*x+3)=1+√(,x+1)

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

(√(,2*x+3))2=(1+√(,x+1))2

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

2*x+3=1+2√(,x+1)+(x+1)

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

2*x+3=x+2+2√(,x+1)

5. Isolate the remaining radical term by subtracting x and 2 from both sides.

x+1=2√(,x+1)

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

(x+1)2=(2√(,x+1))2

7. Expand and simplify both sides of the equation.

x2+2*x+1=4*(x+1)

x2+2*x+1=4*x+4

8. Rearrange the equation into standard quadratic form a*x2+b*x+c=0 by subtracting 4*x and 4 from both sides.

x2−2*x−3=0

9. Factor the quadratic expression by finding two numbers that multiply to −3 and add to −2

(x−3)*(x+1)=0

10. Solve for x by setting each factor equal to zero.

x−3=0⇒x=3

x+1=0⇒x=−1

11. Verify the solutions in the original equation to check for extraneous roots. For x=3 √(,9)−√(,4)=3−2=1 (True). For x=−1 √(,1)−√(,0)=1−0=1 (True).

Final Answer

√(,2*x+3)−√(,x+1)=1⇒x=3,−1

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