To solve the quadratic equation [tex]\( x^2 + x - 42 = 0 \)[/tex], we will follow these steps:
1. Identify the coefficients: The given quadratic equation is in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
[tex]\[
a = 1, \quad b = 1, \quad c = -42
\][/tex]
2. Calculate the discriminant: The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 1^2 - 4 \cdot 1 \cdot (-42)
\][/tex]
Simplifying within the equation:
[tex]\[
\Delta = 1 + 168 = 169
\][/tex]
3. Find the roots using the quadratic formula: The roots of the quadratic equation are given by the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Plugging in [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( \Delta = 169 \)[/tex]:
[tex]\[
x = \frac{-1 \pm \sqrt{169}}{2 \cdot 1}
\][/tex]
Since [tex]\( \sqrt{169} = 13 \)[/tex]:
[tex]\[
x = \frac{-1 \pm 13}{2}
\][/tex]
4. Calculate the individual roots:
[tex]\[
x_1 = \frac{-1 + 13}{2} = \frac{12}{2} = 6
\][/tex]
[tex]\[
x_2 = \frac{-1 - 13}{2} = \frac{-14}{2} = -7
\][/tex]
Hence, the roots of the equation [tex]\( x^2 + x - 42 = 0 \)[/tex] are:
[tex]\[
x_1 = 6 \quad \text{and} \quad x_2 = -7
\][/tex]
To summarize:
- The discriminant [tex]\( \Delta \)[/tex] is 169.
- The roots of the quadratic equation are [tex]\( x_1 = 6 \)[/tex] and [tex]\( x_2 = -7 \)[/tex].
