To find the value of [tex]\( k \)[/tex] such that the roots of the quadratic equation [tex]\( x^2 + kx + 16 = 0 \)[/tex] are equal, we can follow these steps:
1. Understand the condition for equal roots:
- For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the roots are equal if and only if the discriminant is zero.
- The discriminant ([tex]\( \Delta \)[/tex]) of the quadratic equation is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
2. Identify the coefficients:
- In the given equation [tex]\( x^2 + kx + 16 = 0 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = k \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = 16 \)[/tex] (constant term)
3. Formulate the discriminant:
- Use the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] in the discriminant formula:
[tex]\[
\Delta = k^2 - 4 \cdot 1 \cdot 16
\][/tex]
Simplifying, we get:
[tex]\[
\Delta = k^2 - 64
\][/tex]
4. Set the discriminant to zero:
- For the roots to be equal, the discriminant must be zero:
[tex]\[
k^2 - 64 = 0
\][/tex]
5. Solve for [tex]\( k \)[/tex]:
- Solve the equation [tex]\( k^2 - 64 = 0 \)[/tex] for [tex]\( k \)[/tex]:
[tex]\[
k^2 = 64
\][/tex]
[tex]\[
k = \pm 8
\][/tex]
6. Conclude the values of [tex]\( k \)[/tex]:
- The values of [tex]\( k \)[/tex] that make the roots of the equation [tex]\( x^2 + kx + 16 = 0 \)[/tex] equal are:
[tex]\[
k = -8 \quad \text{and} \quad k = 8
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] such that the roots of the equation [tex]\( x^2 + kx + 16 = 0 \)[/tex] are equal are [tex]\( k = -8 \)[/tex] and [tex]\( k = 8 \)[/tex].
