To solve the quadratic equation [tex]\( x^2 + 6x - 5 = 0 \)[/tex], we follow the standard steps used for solving quadratic equations of the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
1. Identify the coefficients:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- [tex]\( c = -5 \)[/tex]
2. Calculate the discriminant using the formula [tex]\(\Delta = b^2 - 4ac\)[/tex]:
[tex]\[
\Delta = 6^2 - 4 \times 1 \times -5 = 36 + 20 = 56
\][/tex]
3. Find the two solutions of the quadratic equation using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex]:
- For the positive root [tex]\(\left(x_1\right)\)[/tex]:
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-6 + \sqrt{56}}{2 \times 1} = \frac{-6 + \sqrt{56}}{2}
\][/tex]
On calculating the above value, we get:
[tex]\[
x_1 \approx 0.7416573867739413
\][/tex]
- For the negative root [tex]\(\left(x_2\right)\)[/tex]:
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-6 - \sqrt{56}}{2 \times 1} = \frac{-6 - \sqrt{56}}{2}
\][/tex]
On calculating the above value, we get:
[tex]\[
x_2 \approx -6.741657386773941
\][/tex]
Thus, the solutions to the quadratic equation [tex]\( x^2 + 6x - 5 = 0 \)[/tex] are:
[tex]\[
x_1 \approx 0.7416573867739413 \quad \text{and} \quad x_2 \approx -6.741657386773941
\][/tex]
So the solutions can be written as:
[tex]\[
x = m \pm n
\][/tex]
Where [tex]\( m \)[/tex] and [tex]\( n \)[/tex] correspond to the approximate values:
- [tex]\( x = 0.7416573867739413 \)[/tex]
- [tex]\( x = -6.741657386773941 \)[/tex]
Replacing these values, the solutions of the equation [tex]\(x^2 + 6x - 5 = 0 \)[/tex] are:
[tex]\[
x = 0.7416573867739413 \quad \text{and} \quad x = -6.741657386773941
\][/tex]
