Problem 133 - Algebra and Trigonometry

Question

Solve: \(\sqrt{6 x-2}=\sqrt{2 x+3}-\sqrt{4 x-1}\)

Step-by-Step Solution

Answer

Solving the equation will yield two possible solutions. But these need to be checked back in the original equation to see which one(s) hold true.

Step 1: Rearrangement of Terms

Firstly, let's isolate \( \sqrt{6x-2} \) by moving the other two square root expressions to the other side of the equation: \( \sqrt{6x-2} + \sqrt{4x-1} = \sqrt{2x+3}. \)

Step 2: Squaring Both Sides of the Equation

Now we square both sides to remove the square root: \( (\sqrt{6x-2} + \sqrt{4x-1})^2 = (\sqrt{2x+3})^2 \). After squaring, we'll get a quadratic equation.

Step 3: Expanding and Simplifying

Let's expand the left hand side and simplify the right hand side: \( (6x - 2 + 2 \sqrt{(6x - 2)(4x - 1)} + 4x - 1) = 2x + 3 \). Now we collect like terms: \( 4 \sqrt{(6x-2)(4x-1)} = 2 - 8x \)

Step 4: Squaring Again to Remove the Square Root

We square both sides again to remove the square root: \( 16(6x - 2)(4x - 1) = (2 - 8x)^2 \). This gives us another quadratic equation.

Step 5: Further Expansion and Simplification

Let's expand and simplify until we get a quadratic equation: after squaring, simplifying and grouping like terms, we'll get a quadratic equation in standard form.

Step 6: Solving the Quadratic Equation

Now, we solve the quadratic equation we get in Step 5 using quadratic formula or completing the squares method. Here, we should also double check if our solutions are true for the original equation since transformations applied might have introduced extraneous solutions.