To solve the quadratic equation [tex]\( p^2 + 11p - 12 = 0 \)[/tex] using the quadratic formula, follow these steps:
1. Identify the coefficients:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In our equation [tex]\( p^2 + 11p - 12 = 0 \)[/tex], the coefficients are:
[tex]\[
a = 1, \quad b = 11, \quad c = -12
\][/tex]
2. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is the part of the quadratic formula under the square root:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 11^2 - 4 \times 1 \times -12
\][/tex]
Calculate [tex]\(11^2 = 121\)[/tex] and [tex]\(4 \times 1 \times -12 = -48\)[/tex], thus:
[tex]\[
\Delta = 121 + 48 = 169
\][/tex]
So, the discriminant [tex]\(\Delta = 169\)[/tex].
3. Calculate the two solutions using the quadratic formula:
The quadratic formula gives us two potential solutions, using the [tex]\(\pm\)[/tex] sign. We need to compute both:
[tex]\[
p = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values [tex]\(b = 11\)[/tex], [tex]\(\Delta = 169\)[/tex], and [tex]\(a = 1\)[/tex]:
[tex]\[
p = \frac{-11 \pm \sqrt{169}}{2 \times 1}
\][/tex]
Calculate [tex]\(\sqrt{169} = 13\)[/tex]:
[tex]\[
p = \frac{-11 \pm 13}{2}
\][/tex]
4. Find the individual solutions:
Splitting into two cases, we have:
Case 1 ([tex]\(+\)[/tex] sign):
[tex]\[
p_1 = \frac{-11 + 13}{2} = \frac{2}{2} = 1
\][/tex]
Case 2 ([tex]\(-\)[/tex] sign):
[tex]\[
p_2 = \frac{-11 - 13}{2} = \frac{-24}{2} = -12
\][/tex]
So, the two solutions to the quadratic equation [tex]\( p^2 + 11p - 12 = 0 \)[/tex] are:
[tex]\[
p_1 = 1 \quad \text{and} \quad p_2 = -12
\][/tex]
In summary:
- The discriminant is [tex]\( \Delta = 169 \)[/tex].
- The solutions are [tex]\( p_1 = 1 \)[/tex] and [tex]\( p_2 = -12 \)[/tex].
