To solve the quadratic equation [tex]\( s^2 - s - 56 = 0 \)[/tex], we follow these steps:
1. Identify the coefficients: In the given quadratic equation [tex]\( s^2 - s - 56 = 0 \)[/tex],
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( s^2 \)[/tex])
- [tex]\( b = -1 \)[/tex] (coefficient of [tex]\( s \)[/tex])
- [tex]\( c = -56 \)[/tex] (constant term)
2. Quadratic Formula: The quadratic formula is given by:
[tex]\[
s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant: The discriminant ([tex]\( \Delta \)[/tex]) is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = (-1)^2 - 4 \cdot 1 \cdot (-56) = 1 + 224 = 225
\][/tex]
4. Find the square root of the discriminant: Since the discriminant is 225, we have:
[tex]\[
\sqrt{225} = 15
\][/tex]
5. Apply the quadratic formula: Substituting [tex]\( b = -1 \)[/tex], [tex]\( a = 1 \)[/tex], and [tex]\( \sqrt{\Delta} = 15 \)[/tex] into the quadratic formula:
[tex]\[
s = \frac{-(-1) \pm 15}{2 \cdot 1} = \frac{1 \pm 15}{2}
\][/tex]
6. Calculate the solutions: This gives us two possible values for [tex]\( s \)[/tex]:
[tex]\[
s_1 = \frac{1 + 15}{2} = \frac{16}{2} = 8
\][/tex]
[tex]\[
s_2 = \frac{1 - 15}{2} = \frac{-14}{2} = -7
\][/tex]
Therefore, the solutions to the equation [tex]\( s^2 - s - 56 = 0 \)[/tex] are:
[tex]\[
s = 8, -7
\][/tex]
So, the solutions are [tex]\( s = 8, -7 \)[/tex].
