The common root of the equations `x^(2)-7x+10=0` and `x^(2)-10x+16=0` is

Options

A. `-2`
B. `3`
C. 5
D. 2

Answer

To find the common root of the equations \( x^2 - 7x + 10 = 0 \) and \( x^2 - 10x + 16 = 0 \), we will solve both equations step by step.

### Step 1: Solve the first equation \( x^2 - 7x + 10 = 0 \)

We can factor this quadratic equation. We need two numbers that multiply to \( 10 \) (the constant term) and add up to \( -7 \) (the coefficient of \( x \)).

The factors of \( 10 \) that satisfy these conditions are \( -5 \) and \( -2 \).

Thus, we can write:
\[ x^2 - 7x + 10 = (x - 5)(x - 2) = 0 \]

### Step 2: Find the roots of the first equation

Setting each factor to zero gives us:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

So, the roots of the first equation are \( x = 5 \) and \( x = 2 \).

### Step 3: Solve the second equation \( x^2 - 10x + 16 = 0 \)

Similarly, we will factor this quadratic equation. We need two numbers that multiply to \( 16 \) and add up to \( -10 \).

The factors of \( 16 \) that satisfy these conditions are \( -2 \) and \( -8 \).

Thus, we can write:
\[ x^2 - 10x + 16 = (x - 2)(x - 8) = 0 \]

### Step 4: Find the roots of the second equation

Setting each factor to zero gives us:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \]

So, the roots of the second equation are \( x = 2 \) and \( x = 8 \).

### Step 5: Identify the common root

Now we compare the roots of both equations:
- Roots of the first equation: \( 2, 5 \)
- Roots of the second equation: \( 2, 8 \)

The common root between the two equations is \( x = 2 \).

### Final Answer

The common root of the equations \( x^2 - 7x + 10 = 0 \) and \( x^2 - 10x + 16 = 0 \) is \( \boxed{2} \).

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