Find the square root of
`(i) (3//2)(x-1)+sqrt(2x^(2)-7x-4)`
`(ii) 1-x+sqrt(22x-15-8x^(2))`.

Answer

To solve the given problems, we will find the square root of the expressions step by step.

### Part (i): Find the square root of \( \frac{3}{2}(x - 1) + \sqrt{2x^2 - 7x - 4} \)

**Step 1: Simplify the expression inside the square root.**

We start with:
\[ \frac{3}{2}(x - 1) + \sqrt{2x^2 - 7x - 4} \]

**Step 2: Factor the quadratic expression.**

We need to factor \(2x^2 - 7x - 4\). We look for two numbers that multiply to \(2 \cdot (-4) = -8\) and add to \(-7\). The numbers are \(-8\) and \(1\).

Rewriting the expression:
\[ 2x^2 - 8x + x - 4 = 2x(x - 4) + 1(x - 4) = (2x + 1)(x - 4) \]

So, we have:
\[ \sqrt{2x^2 - 7x - 4} = \sqrt{(2x + 1)(x - 4)} \]

**Step 3: Substitute back into the expression.**

Now, substituting back, we have:
\[ \frac{3}{2}(x - 1) + \sqrt{(2x + 1)(x - 4)} \]

**Step 4: Combine the terms.**

Now we can express the entire expression as:
\[ \frac{3}{2}(x - 1) + \sqrt{(2x + 1)(x - 4)} \]

**Step 5: Rewrite the expression in a perfect square form.**

We can rewrite this as:
\[ \sqrt{\left(\sqrt{\frac{3}{2}(x - 1)} + \sqrt{(2x + 1)(x - 4)}\right)^2} \]

**Step 6: Final expression.**

Thus, the square root is:
\[ \sqrt{2} \left( \sqrt{\frac{x - 4}{2}} + \sqrt{\frac{2x + 1}{2}} \right) \]

### Part (ii): Find the square root of \( 1 - x + \sqrt{22x - 15 - 8x^2} \)

**Step 1: Simplify the expression inside the square root.**

We start with:
\[ 1 - x + \sqrt{22x - 15 - 8x^2} \]

**Step 2: Rearrange the quadratic expression.**

Rearranging the quadratic:
\[ -8x^2 + 22x - 15 \]
This can be rewritten as:
\[ -8(x^2 - \frac{22}{8}x + \frac{15}{8}) \]

**Step 3: Factor the quadratic expression.**

We need to factor \(x^2 - \frac{22}{8}x + \frac{15}{8}\). The numbers that multiply to \(\frac{15}{8}\) and add to \(-\frac{22}{8}\) are \(-\frac{15}{4}\) and \(-\frac{3}{2}\).

So we can write:
\[ \sqrt{-8\left(x - \frac{15}{4}\right)\left(x - \frac{3}{2}\right)} \]

**Step 4: Substitute back into the expression.**

Now substituting back, we have:
\[ 1 - x + \sqrt{-8\left(x - \frac{15}{4}\right)\left(x - \frac{3}{2}\right)} \]

**Step 5: Rewrite the expression in a perfect square form.**

We can express this as:
\[ \sqrt{\left(\sqrt{1 - x} + \sqrt{-8\left(x - \frac{15}{4}\right)\left(x - \frac{3}{2}\right)}\right)^2} \]

**Step 6: Final expression.**

Thus, the square root is:
\[ \sqrt{2} \left( \sqrt{1 - x} + \sqrt{-8\left(x - \frac{15}{4}\right)\left(x - \frac{3}{2}\right)} \right) \]