Solve the equation:
_ ___
�ãx +2=�ã5-x +3
illegible symbols
You later posted this as :
sqrt(x) + 2 = sqrt (5-x) + 3
The answers are x = 1 and 4, and depend upon whether you take positive or negative square roots, or one of each.
Look for my derivation in the other place where you posted this.
To solve the equation √(x+2) = √(5-x) + 3, we need to follow a few steps:
Step 1: Square both sides of the equation to eliminate the square roots.
(√(x+2))^2 = (√(5-x) + 3)^2
Simplifying, we get:
x + 2 = (5 - x) + 2√(5 - x) + 9
Step 2: Combine like terms.
x + 2 = 5 - x + 2√(5 - x) + 9
Rearranging the terms:
x + x + 2√(5 - x) = 5 + 9 - 2
Simplifying:
2x + 2√(5 - x) = 12
Step 3: Move constant terms to the other side of the equation.
2√(5 - x) = 12 - 2x
Step 4: Divide by 2 to isolate the square root term.
√(5 - x) = (12 - 2x) / 2
√(5 - x) = 6 - x
Step 5: Square both sides again to eliminate the square root.
(√(5 - x))^2 = (6 - x)^2
Simplifying:
5 - x = 36 - 12x + x^2
Step 6: Rearrange the equation to set it equal to zero.
0 = x^2 - 11x + 31
Step 7: Solve the quadratic equation by factoring or using the quadratic formula.
If you factor the equation, you will find that it cannot be factored easily. So, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -11, and c = 31. Substituting these values into the formula:
x = (-(-11) ± √((-11)^2 - 4(1)(31))) / (2(1))
x = (11 ± √(121 - 124)) / 2
x = (11 ± √(-3)) / 2
Since we have a negative value under the square root, this equation has no real solutions. Therefore, there is no real number that satisfies the original equation √(x+2) = √(5-x) + 3.