The values of a for which `y=x^2+ax+25` touches the axis of x are:

Options

A. `+-5`
B. `+-10`
C. `+-2`
D. `+-1`

Answer

To find the values of \( a \) for which the quadratic equation \( y = x^2 + ax + 25 \) touches the x-axis, we need to ensure that the equation has exactly one solution. This occurs when the discriminant of the quadratic equation is zero.

### Step-by-Step Solution:

1. **Identify the quadratic equation**:
The given equation is:
\[ y = x^2 + ax + 25 \]

2. **Set the equation to zero**:
To find the points where the curve touches the x-axis, we set \( y = 0 \):
\[ x^2 + ax + 25 = 0 \]

3. **Calculate the discriminant**:
The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
Here, \( A = 1 \), \( B = a \), and \( C = 25 \). Thus, the discriminant is:
\[ D = a^2 - 4 \cdot 1 \cdot 25 = a^2 - 100 \]

4. **Set the discriminant to zero**:
For the quadratic to touch the x-axis, the discriminant must be zero:
\[ a^2 - 100 = 0 \]

5. **Solve for \( a \)**:
Rearranging the equation gives:
\[ a^2 = 100 \]
Taking the square root of both sides results in:
\[ a = \pm 10 \]

6. **Conclusion**:
The values of \( a \) for which the quadratic touches the x-axis are:
\[ a = 10 \quad \text{or} \quad a = -10 \]

### Final Answer:
The values of \( a \) for which \( y = x^2 + ax + 25 \) touches the x-axis are \( a = 10 \) and \( a = -10 \).

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