If `x = k^2` and y = k is a solution of x - 5y + 6 = 0, then find the values of k.

Options

A. 1,2
B. 2,3
C. 1,5
D. 2,4

Answer

To solve the problem, we start with the given equations and substitute the values of \( x \) and \( y \).

### Step 1: Substitute the values of \( x \) and \( y \)
We know:
- \( x = k^2 \)
- \( y = k \)

We substitute these into the equation \( x - 5y + 6 = 0 \):
\[ k^2 - 5(k) + 6 = 0 \]

### Step 2: Simplify the equation
This simplifies to:
\[ k^2 - 5k + 6 = 0 \]

### Step 3: Factor the quadratic equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to \( 6 \) (the constant term) and add up to \( -5 \) (the coefficient of \( k \)). These numbers are \( -2 \) and \( -3 \).

Thus, we can factor the equation as:
\[ (k - 2)(k - 3) = 0 \]

### Step 4: Solve for \( k \)
Now, we set each factor equal to zero:
1. \( k - 2 = 0 \) → \( k = 2 \)
2. \( k - 3 = 0 \) → \( k = 3 \)

### Conclusion
The values of \( k \) are:
\[ k = 2 \quad \text{and} \quad k = 3 \]

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