Solve`sqrt(3x^2 -7x -30) - sqrt(2 x^2 -7x-5) = x-5`

Options

A. one
B. two
C. three
D. none of these

Answer

To solve the equation \( \sqrt{3x^2 - 7x - 30} - \sqrt{2x^2 - 7x - 5} = x - 5 \), we will follow these steps:

### Step 1: Isolate one of the square roots
We can rewrite the equation by isolating one of the square roots:
\[ \sqrt{3x^2 - 7x - 30} = x - 5 + \sqrt{2x^2 - 7x - 5} \]

**Hint:** Always isolate one square root before squaring to simplify the equation.

### Step 2: Square both sides
Next, we square both sides to eliminate the square root:
\[ 3x^2 - 7x - 30 = (x - 5 + \sqrt{2x^2 - 7x - 5})^2 \]
Expanding the right side:
\[ = (x - 5)^2 + 2(x - 5)\sqrt{2x^2 - 7x - 5} + (2x^2 - 7x - 5) \]
\[ = (x^2 - 10x + 25) + 2(x - 5)\sqrt{2x^2 - 7x - 5} + (2x^2 - 7x - 5) \]

**Hint:** When squaring, remember to apply the binomial expansion correctly.

### Step 3: Combine like terms
Now, we combine like terms:
\[ 3x^2 - 7x - 30 = x^2 - 10x + 25 + 2x^2 - 7x - 5 + 2(x - 5)\sqrt{2x^2 - 7x - 5} \]
This simplifies to:
\[ 3x^2 - 7x - 30 = 3x^2 - 17x + 20 + 2(x - 5)\sqrt{2x^2 - 7x - 5} \]

**Hint:** Keep track of all terms when combining like terms to avoid mistakes.

### Step 4: Move terms to one side
Now, we can move all terms to one side:
\[ 3x^2 - 7x - 30 - (3x^2 - 17x + 20) = 2(x - 5)\sqrt{2x^2 - 7x - 5} \]
This simplifies to:
\[ 10x - 50 = 2(x - 5)\sqrt{2x^2 - 7x - 5} \]

**Hint:** Moving all terms to one side helps in isolating the remaining square root.

### Step 5: Isolate the square root
We can further simplify:
\[ 5(x - 5) = (x - 5)\sqrt{2x^2 - 7x - 5} \]
Assuming \( x - 5 \neq 0 \), we can divide both sides by \( x - 5 \):
\[ 5 = \sqrt{2x^2 - 7x - 5} \]

**Hint:** Ensure that you check for any restrictions when dividing by a variable expression.

### Step 6: Square both sides again
Now, square both sides again:
\[ 25 = 2x^2 - 7x - 5 \]
Rearranging gives:
\[ 2x^2 - 7x - 30 = 0 \]

**Hint:** After squaring, always rearrange to form a standard quadratic equation.

### Step 7: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2, b = -7, c = -30 \):
\[ x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 2 \cdot (-30)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ = \frac{7 \pm \sqrt{49 + 240}}{4} = \frac{7 \pm \sqrt{289}}{4} = \frac{7 \pm 17}{4} \]
This gives us two solutions:
\[ x = \frac{24}{4} = 6 \quad \text{and} \quad x = \frac{-10}{4} = -\frac{5}{2} \]

**Hint:** Always check your solutions in the original equation to ensure they are valid.

### Step 8: Verify the solutions
Substituting \( x = 6 \) and \( x = -\frac{5}{2} \) back into the original equation to check if they satisfy it.

After checking, we find that \( x = 6 \) is a valid solution, while \( x = -\frac{5}{2} \) does not satisfy the original equation.

### Final Answer
The solution to the equation is:
\[ \boxed{6} \]