The given quadratic equation is [tex]\(7x^2 + 19x - 6 = 0\)[/tex]. To find the solutions for [tex]\( x \)[/tex], we can use the quadratic formula, which is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 19 \)[/tex]
- [tex]\( c = -6 \)[/tex]
The quadratic formula involves calculating the discriminant ([tex]\(\Delta\)[/tex]) first, which is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
The solutions for [tex]\( x \)[/tex] are then given by:
[tex]\[ x = \frac{-b + \sqrt{\Delta}}{2a} \quad \text{and} \quad x = \frac{-b - \sqrt{\Delta}}{2a} \][/tex]
Given that the discriminant ([tex]\(\Delta\)[/tex]) and the entire solution have already been calculated, we find that the solutions are:
- Smaller solution: [tex]\( x = -3.0 \)[/tex]
- Larger solution: [tex]\( x = 0.2857142857142857 \)[/tex]
Thus, the two solutions to the equation [tex]\(7x^2 + 19x - 6 = 0\)[/tex] are:
Smaller solution is [tex]\( x = -3.0 \)[/tex]
Larger solution is [tex]\( x = 0.2857142857142857 \)[/tex]
