The values ·of a for whic.h `y = x^2 +ax+ 25` touches the axis of x are

Options

A. `pm5`
B. `pm10`
C. `pm15`
D. none

Answer

To find the values of \( a \) for which the curve \( y = x^2 + ax + 25 \) touches the x-axis, we can follow these steps:

### Step 1: Set the equation to zero
Since the curve touches the x-axis, we set \( y = 0 \):
\[ 0 = x^2 + ax + 25 \]
This is our equation (1).

### Step 2: Condition for tangency
For the curve to touch the x-axis, the quadratic equation must have exactly one solution. This occurs when the discriminant is zero. The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
In our case, \( A = 1 \), \( B = a \), and \( C = 25 \). Thus, the discriminant becomes:
\[ D = a^2 - 4(1)(25) = a^2 - 100 \]

### Step 3: Set the discriminant to zero
For the curve to touch the x-axis, we set the discriminant to zero:
\[ a^2 - 100 = 0 \]

### Step 4: Solve for \( a \)
Now, we solve the equation:
\[ a^2 = 100 \]
Taking the square root of both sides gives:
\[ a = \pm 10 \]

### Conclusion
The values of \( a \) for which the curve \( y = x^2 + ax + 25 \) touches the x-axis are:
\[ a = 10 \quad \text{and} \quad a = -10 \]