Number of values of `x` satisfying the pair of quadratic equations `x^(2)-px+20=0` and `x^(2)-20x+p=0` for some `p in R` is

Options

A. `0`
B. `1`
C. `2`
D. `3`

Explanation

To find the number of values of \( x \) satisfying the pair of quadratic equations

1. \( x^2 - px + 20 = 0 \)
2. \( x^2 - 20x + p = 0 \)

for some \( p \in \mathbb{R} \), we can analyze the equations step by step.

### Step 1: Analyze the equations

The two equations are:

1. \( x^2 - px + 20 = 0 \) (Equation 1)
2. \( x^2 - 20x + p = 0 \) (Equation 2)

### Step 2: Set the equations equal

To find the relationship between \( p \) and \( x \), we can subtract Equation 1 from Equation 2:

\[ (x^2 - 20x + p) - (x^2 - px + 20) = 0 \]

This simplifies to:

\[ -p + px - 20 + p = 0 \]

Which further simplifies to:

\[ (px - 20) = 0 \]

### Step 3: Solve for \( p \)

From \( px - 20 = 0 \), we can express \( p \) in terms of \( x \):

\[ p = \frac{20}{x} \quad (x \neq 0) \]

### Step 4: Substitute \( p \) back into one of the equations

Now, we can substitute \( p \) back into either of the original equations. Let's substitute into Equation 1:

\[ x^2 - \left(\frac{20}{x}\right)x + 20 = 0 \]

This simplifies to:

\[ x^2 - 20 + 20 = 0 \]

Thus, we have:

\[ x^2 = 0 \]

### Step 5: Solve for \( x \)

Taking the square root of both sides gives:

\[ x = 0 \]

### Step 6: Consider the case when \( p = 20 \)

If we set \( p = 20 \), both equations become identical:

\[ x^2 - 20x + 20 = 0 \]

We can find the roots of this equation using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{20 \pm \sqrt{20^2 - 4 \cdot 1 \cdot 20}}{2 \cdot 1} \]

Calculating the discriminant:

\[ 20^2 - 80 = 400 - 80 = 320 \]

Thus, the roots are:

\[ x = \frac{20 \pm \sqrt{320}}{2} = \frac{20 \pm 8\sqrt{5}}{2} = 10 \pm 4\sqrt{5} \]

### Conclusion

Now we have two cases:

1. When \( p \) is any real number except 20, we have one solution \( x = 0 \).
2. When \( p = 20 \), we have two solutions \( x = 10 + 4\sqrt{5} \) and \( x = 10 - 4\sqrt{5} \).

Thus, the total number of distinct values of \( x \) satisfying the equations is:

- 1 value when \( p \neq 20 \)
- 2 values when \( p = 20 \)

The total number of values of \( x \) satisfying the equations is **2**.