Solve for x 2x-5 = square root of x+4

Problem

2*x−5=√(,x+4)

Solution

1. Square both sides of the equation to eliminate the square root on the right side.

(2*x−5)2=(√(,x+4))2

2. Expand the left side using the square of a binomial formula (a−b)2=a2−2*a*b+b2

4*x2−20*x+25=x+4

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

4*x2−21*x+21=0

4. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=4 b=−21 and c=21

x=(−(−21)±√(,(−21)2−4*(4)*(21)))/(2*(4))

5. Simplify the discriminant and the rest of the expression.

x=(21±√(,441−336))/8

x=(21±√(,105))/8

6. Check for extraneous solutions by substituting the values back into the original equation. Since √(,105)≈10.25 the value x=(21−10.25)/8≈1.34 would make the left side 2(1.34) - 5$ negative, while the square root must be non-negative. Thus, only the positive root is valid.

2*((21+√(,105))/8)−5=√(,(21+√(,105))/8+4)

Final Answer

x=(21+√(,105))/8

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