Solve: sqrt(7x-5)+sqrt(15x+4)=sqrt(30x+31)
To solve the given equation: √(7x-5) + √(15x+4) = √(30x+31), we need to isolate the variable x.
Step 1: Start by subtracting √(7x-5) from both sides of the equation:
√(15x+4) = √(30x+31) - √(7x-5)
Step 2: To eliminate the square root on the left side, we need to square both sides of the equation:
(√(15x+4))^2 = (√(30x+31) - √(7x-5))^2
Simplifying the left side:
15x+4 = (30x+31) - 2√((30x+31)(7x-5)) + (7x-5)
Step 3: Group the like terms and isolate the radical term:
15x+4 - 33x + 5 = 2√((30x+31)(7x-5))
-18x + 9 = 2√((30x+31)(7x-5))
Step 4: Square both sides again to eliminate the radical:
(-18x + 9)^2 = (2√((30x+31)(7x-5)))^2
Simplifying the left side:
324x^2 - 324x + 81 = 4((30x+31)(7x-5))
Step 5: Expand and simplify the right side:
324x^2 - 324x + 81 = 4(210x^2 - 330x + 217x -155)
324x^2 - 324x + 81 = 840x^2 - 1320x + 868x - 620
324x^2 - 324x + 81 = 840x^2 - 452x - 620
Step 6: Combine like terms and rewrite the equation in standard form:
840x^2 - 324x^2 - 324x - 452x - 620 + 81 = 0
516x^2 - 776x - 539 = 0
Now, we have a quadratic equation in standard form. We can solve it using various methods such as factoring, completing the square, or using the quadratic formula.
Note: Factoring this particular quadratic equation might not be easy, so we will proceed with the quadratic formula.
The quadratic formula is given by: x = [-b ± √(b^2 - 4ac)] / 2a
For the equation 516x^2 - 776x - 539 = 0:
a = 516
b = -776
c = -539
Using the quadratic formula:
x = [-(-776) ± √((-776)^2 - 4 * 516 * -539)] / 2 * 516
x = [776 ± √(602176 + 1110192)] / 1032
x = [776 ± √(1712368)] / 1032
x = [776 ± 1308.833989] / 1032
Simplifying further:
x = (776 + 1308.833989) / 1032 or x = (776 - 1308.833989) / 1032
x = 2.0919 or x = -0.4975
So, the solutions to the equation are x = 2.0919 and x = -0.4975.