Solve for x: (i) `10x^(2)-27x+5=0` (ii) `pqx^(2)-(p^(2)+q^(2))x+pq=0`

Answer

To solve the given equations step by step, we will tackle each equation one at a time.

### (i) Solve for x: \(10x^2 - 27x + 5 = 0\)

**Step 1: Rewrite the equation**
We start with the equation:
\[ 10x^2 - 27x + 5 = 0 \]

**Step 2: Factor the quadratic**
We need to factor the quadratic expression. To do this, we can look for two numbers that multiply to \(10 \times 5 = 50\) and add up to \(-27\). We can rewrite the middle term:
\[ 10x^2 - 25x - 2x + 5 = 0 \]

**Step 3: Group the terms**
Now, we group the terms:
\[ (10x^2 - 25x) + (-2x + 5) = 0 \]

**Step 4: Factor by grouping**
Factoring out common terms from each group:
\[ 5x(2x - 5) - 1(2x - 5) = 0 \]
Now, we can factor out \((2x - 5)\):
\[ (2x - 5)(5x - 1) = 0 \]

**Step 5: Set each factor to zero**
Now we set each factor to zero:
1. \(2x - 5 = 0\)
2. \(5x - 1 = 0\)

**Step 6: Solve for x**
From \(2x - 5 = 0\):
\[ 2x = 5 \implies x = \frac{5}{2} \]
From \(5x - 1 = 0\):
\[ 5x = 1 \implies x = \frac{1}{5} \]

**Final Solutions:**
The solutions for the equation \(10x^2 - 27x + 5 = 0\) are:
\[ x = \frac{5}{2} \quad \text{and} \quad x = \frac{1}{5} \]

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### (ii) Solve for x: \(pqx^2 - (p^2 + q^2)x + pq = 0\)

**Step 1: Rewrite the equation**
We start with the equation:
\[ pqx^2 - (p^2 + q^2)x + pq = 0 \]

**Step 2: Divide by pq**
To simplify, we can divide the entire equation by \(pq\) (assuming \(pq \neq 0\)):
\[ x^2 - \left(\frac{p^2 + q^2}{pq}\right)x + 1 = 0 \]

**Step 3: Identify coefficients**
In the standard form \(ax^2 + bx + c = 0\), we have:
- \(a = 1\)
- \(b = -\left(\frac{p^2 + q^2}{pq}\right)\)
- \(c = 1\)

**Step 4: Use the quadratic formula**
We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{\left(\frac{p^2 + q^2}{pq}\right) \pm \sqrt{\left(\frac{p^2 + q^2}{pq}\right)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]

**Step 5: Simplify the expression**
Calculating the discriminant:
\[ \left(\frac{p^2 + q^2}{pq}\right)^2 - 4 = \frac{(p^2 + q^2)^2}{p^2q^2} - 4 \]
This can be rewritten as:
\[ \frac{(p^2 + q^2)^2 - 4p^2q^2}{p^2q^2} \]

**Step 6: Find the roots**
Now we can express the roots:
\[ x = \frac{\frac{p^2 + q^2}{pq} \pm \sqrt{\frac{(p^2 + q^2)^2 - 4p^2q^2}{p^2q^2}}}{2} \]
This simplifies to:
\[ x = \frac{p^2 + q^2 \pm \sqrt{(p^2 - q^2)^2}}{2pq} \]
Thus, the roots are:
\[ x = \frac{p^2 + q^2 + (p^2 - q^2)}{2pq} = \frac{2p^2}{2pq} = \frac{p}{q} \quad \text{and} \quad x = \frac{p^2 + q^2 - (p^2 - q^2)}{2pq} = \frac{2q^2}{2pq} = \frac{q}{p} \]

**Final Solutions:**
The solutions for the equation \(pqx^2 - (p^2 + q^2)x + pq = 0\) are:
\[ x = \frac{p}{q} \quad \text{and} \quad x = \frac{q}{p} \]

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