If `x ^(2) - hx - 21 =0, x ^(2) - 3 hx + 35 =0 (h gt 0)` has a common root, then the value of h is equal to

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the value of \( h \) such that the equations \( x^2 - hx - 21 = 0 \) and \( x^2 - 3hx + 35 = 0 \) have a common root, we can follow these steps:

### Step 1: Assume a common root
Let \( a \) be the common root of both equations.

### Step 2: Substitute the common root into the first equation
Substituting \( a \) into the first equation:
\[ a^2 - ha - 21 = 0 \]
This can be rearranged to:
\[ a^2 = ha + 21 \quad \text{(1)} \]

### Step 3: Substitute the common root into the second equation
Now substitute \( a \) into the second equation:
\[ a^2 - 3ha + 35 = 0 \]
This can be rearranged to:
\[ a^2 = 3ha - 35 \quad \text{(2)} \]

### Step 4: Set the two expressions for \( a^2 \) equal to each other
From equations (1) and (2), we can set them equal to each other:
\[ ha + 21 = 3ha - 35 \]

### Step 5: Solve for \( h \)
Rearranging the equation gives:
\[ 21 + 35 = 3ha - ha \]
\[ 56 = 2ha \]
Dividing both sides by \( 2a \) (assuming \( a \neq 0 \)):
\[ h = \frac{56}{2a} = \frac{28}{a} \quad \text{(3)} \]

### Step 6: Substitute \( h \) back into one of the original equations
Now we will substitute \( h \) from equation (3) back into the first equation:
\[ a^2 - \left(\frac{28}{a}\right)a - 21 = 0 \]
This simplifies to:
\[ a^2 - 28 - 21 = 0 \]
\[ a^2 - 49 = 0 \]
Factoring gives:
\[ (a - 7)(a + 7) = 0 \]
Thus, \( a = 7 \) or \( a = -7 \).

### Step 7: Find the value of \( h \)
Using \( a = 7 \) in equation (3):
\[ h = \frac{28}{7} = 4 \]
Using \( a = -7 \) in equation (3):
\[ h = \frac{28}{-7} = -4 \]
Since \( h > 0 \), we take \( h = 4 \).

### Final Answer
The value of \( h \) is \( 4 \).

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