If one root of the equation `(x-1)(7-x)=m` is three times the other, then `m` is equal to

Options

A. `-5`
B. `0`
C. `2`
D. `5`

Explanation

To solve the equation \((x-1)(7-x)=m\) given that one root is three times the other, we can follow these steps:

### Step 1: Expand the equation
First, we need to expand the left-hand side of the equation:

\[ (x-1)(7-x) = 7x - x^2 - 7 + x = -x^2 + 8x - 7 \]

So, we rewrite the equation as:

\[ -x^2 + 8x - 7 = m \]

Rearranging gives us:

\[ -x^2 + 8x - (7 + m) = 0 \]

Multiplying through by -1 to make the leading coefficient positive:

\[ x^2 - 8x + (7 + m) = 0 \]

### Step 2: Identify the roots
Let the roots of the equation be \(\alpha\) and \(3\alpha\). The sum of the roots can be expressed as:

\[ \alpha + 3\alpha = 4\alpha \]

According to Vieta's formulas, the sum of the roots is equal to the coefficient of \(x\) with the opposite sign:

\[ 4\alpha = 8 \implies \alpha = 2 \]

### Step 3: Calculate the product of the roots
The product of the roots is given by:

\[ \alpha \cdot 3\alpha = 3\alpha^2 \]

Using Vieta's formulas again, the product of the roots is also equal to the constant term divided by the leading coefficient:

\[ 3\alpha^2 = 7 + m \]

Substituting \(\alpha = 2\):

\[ 3(2^2) = 7 + m \implies 3 \cdot 4 = 7 + m \implies 12 = 7 + m \]

### Step 4: Solve for \(m\)
Now, we can solve for \(m\):

\[ m = 12 - 7 = 5 \]

Thus, the value of \(m\) is:

\[ \boxed{5} \]

---