Let's solve the quadratic equation [tex]\( p^2 + 13p - 30 = 0 \)[/tex] to find the values of [tex]\( p \)[/tex].
1. Identify the quadratic equation in standard form:
The given equation [tex]\( p^2 + 13p - 30 = 0 \)[/tex] is already in standard form [tex]\( ax^2 + bx + c = 0 \)[/tex] where:
[tex]\[
a = 1, \quad b = 13, \quad c = -30
\][/tex]
2. Use the quadratic formula:
The quadratic formula is:
[tex]\[
p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 13 \)[/tex], and [tex]\( c = -30 \)[/tex]. Substituting these values into the formula gives:
[tex]\[
p = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 1 \cdot (-30)}}{2 \cdot 1}
\][/tex]
3. Calculate the discriminant:
The discriminant is [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
13^2 - 4 \cdot 1 \cdot (-30) = 169 + 120 = 289
\][/tex]
4. Simplify under the square root:
[tex]\[
\sqrt{289} = 17
\][/tex]
5. Substitute back into the quadratic formula:
[tex]\[
p = \frac{-13 \pm 17}{2}
\][/tex]
6. Solve for the two values of [tex]\( p \)[/tex]:
- For the [tex]\( + \)[/tex] part:
[tex]\[
p = \frac{-13 + 17}{2} = \frac{4}{2} = 2
\][/tex]
- For the [tex]\( - \)[/tex] part:
[tex]\[
p = \frac{-13 - 17}{2} = \frac{-30}{2} = -15
\][/tex]
So, the solutions to the equation [tex]\( p^2 + 13p - 30 = 0 \)[/tex] are [tex]\( p = -15 \)[/tex] and [tex]\( p = 2 \)[/tex].
Therefore, the best answer is:
D. [tex]\( p = -15, p = 2 \)[/tex]
