Solve: sqrt(7x-5)+sqrt(15x+4)=sqrt(30x+31)
First square both sides.
(7x-5)+(15x+4)+ 2sqrt[(7x-5)(15x+4)]= 30x+31
22x-1 +2sqrt[(7x-5)(15x+4)]= 30x +31
sqrt[(7x-5)(15x+4)]= 4x + 16
Now square both sides again and solve the resulting quadratic.
Check my math; I tend to be sloppy.
To solve the equation √(7x - 5) + √(15x + 4) = √(30x + 31):
Step 1: Isolate one of the square roots on one side of the equation. Subtracting √(15x + 4) from both sides, we have √(7x - 5) = √(30x + 31) - √(15x + 4).
Step 2: To eliminate the square root in the denominator, multiply the expression by the conjugate of the denominator. The conjugate of a square root expression is found by changing the sign between the two terms (e.g. √a ± √b changes to √a ∓ √b).
Multiplying both sides of the equation by (√(30x + 31) + √(15x + 4)), we get:
(√(7x - 5)) * (√(30x + 31) + √(15x + 4)) = (√(30x + 31) - √(15x + 4)) * (√(30x + 31) + √(15x + 4)).
Step 3: Simplify both sides:
√(7x - 5) * (√(30x + 31) + √(15x + 4)) = (30x + 31) - √((15x + 4) * (30x + 31)).
Step 4: Continue simplifying:
(√(7x - 5) * √(30x + 31)) + (√(7x - 5) * √(15x + 4)) = 30x + 31 - √((30x + 31) * (15x + 4)).
Step 5: Simplify further:
√((7x - 5) * (30x + 31)) + √((7x - 5) * (15x + 4)) = 30x + 31 - √((30x + 31) * (15x + 4)).
Step 6: Apply the rule of multiplication for square roots:
√((7x - 5) * (30x + 31)) + √((7x - 5) * (15x + 4)) = 30x + 31 - √(450x^2 + 465x + 124).
Step 7: Combine like terms on both sides:
√((7x - 5) * (30x + 31)) + √((7x - 5) * (15x + 4)) + √(450x^2 + 465x + 124) = 30x + 31.
Step 8: Square both sides of the equation to eliminate the square roots:
(√((7x - 5) * (30x + 31)) + √((7x - 5) * (15x + 4)) + √(450x^2 + 465x + 124))^2 = (30x + 31)^2.
Step 9: Simplify both sides and solve for x:
(7x - 5) * (30x + 31) + 2√((7x - 5) * (30x + 31)) * (√((7x - 5) * (15x + 4)) + √(450x^2 + 465x + 124)) + (7x - 5) * (15x + 4) + 2√((7x - 5) * (15x + 4)) * √(450x^2 + 465x + 124) + (450x^2 + 465x + 124) = (30x + 31)^2.
This equation can be continued to be simplified and then solved using algebraic methods such as expanding, collecting like terms, and solving the resulting quadratic equation.