Solve the given quadratic equations and mark the correct options based on your answer
I. ` ( 20 % of 225)/( x) = - x + 14`
II. ` 30% of 70 y =y^(2) +90`

Options

A. ` x ge y `
B. ` x gt y `
C. x = y or no relation can be established between x and y
D. ` x le y `

Answer

To solve the given quadratic equations, we will tackle each equation step by step.

### Equation I:
\[ \frac{20\% \text{ of } 225}{x} = -x + 14 \]

**Step 1: Calculate 20% of 225.**
\[ 20\% \text{ of } 225 = \frac{20}{100} \times 225 = \frac{1}{5} \times 225 = 45 \]

**Step 2: Substitute this value into the equation.**
\[ \frac{45}{x} = -x + 14 \]

**Step 3: Multiply both sides by \(x\) to eliminate the fraction (assuming \(x \neq 0\)).**
\[ 45 = -x^2 + 14x \]

**Step 4: Rearrange the equation to standard quadratic form.**
\[ x^2 - 14x + 45 = 0 \]

**Step 5: Factor the quadratic equation.**
To factor \(x^2 - 14x + 45\), we need two numbers that multiply to \(45\) and add to \(-14\). The numbers are \(-9\) and \(-5\).
\[ (x - 9)(x - 5) = 0 \]

**Step 6: Solve for \(x\).**
Setting each factor to zero gives:
\[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \]
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]

### Equation II:
\[ 30\% \text{ of } 70y = y^2 + 90 \]

**Step 1: Calculate 30% of 70y.**
\[ 30\% \text{ of } 70y = \frac{30}{100} \times 70y = \frac{3}{10} \times 70y = 21y \]

**Step 2: Substitute this value into the equation.**
\[ 21y = y^2 + 90 \]

**Step 3: Rearrange the equation to standard quadratic form.**
\[ y^2 - 21y + 90 = 0 \]

**Step 4: Factor the quadratic equation.**
To factor \(y^2 - 21y + 90\), we need two numbers that multiply to \(90\) and add to \(-21\). The numbers are \(-15\) and \(-6\).
\[ (y - 15)(y - 6) = 0 \]

**Step 5: Solve for \(y\).**
Setting each factor to zero gives:
\[ y - 15 = 0 \quad \Rightarrow \quad y = 15 \]
\[ y - 6 = 0 \quad \Rightarrow \quad y = 6 \]

### Summary of Solutions:
- For Equation I, the solutions are \(x = 9\) and \(x = 5\).
- For Equation II, the solutions are \(y = 15\) and \(y = 6\).

### Comparison of \(x\) and \(y\):
- Compare \(x = 9\) with \(y = 15\): \(9 < 15\)
- Compare \(x = 5\) with \(y = 6\): \(5 < 6\)

### Conclusion:
From the comparisons, we can see that in both cases, \(x\) is less than \(y\). Therefore, we cannot establish a direct relationship where \(x\) is greater than \(y\) in all cases.

### Final Answer:
No relation can be established between \(x\) and \(y\), which corresponds to option C.

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