Sure, let's solve the quadratic equation step by step using the quadratic formula.
The given quadratic equation is [tex]\(3x^2 - x - 10 = 0\)[/tex].
For a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], the quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, we have:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -1\)[/tex]
- [tex]\(c = -10\)[/tex]
Let's first calculate the discriminant, [tex]\(\Delta\)[/tex], which is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-1)^2 - 4(3)(-10) \][/tex]
[tex]\[ \Delta = 1 - (-120) \][/tex]
[tex]\[ \Delta = 1 + 120 \][/tex]
[tex]\[ \Delta = 121 \][/tex]
Now, we'll calculate the two solutions using the quadratic formula:
[tex]\[ x_1 = \frac{-b + \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x_2 = \frac{-b - \sqrt{\Delta}}{2a} \][/tex]
Substituting the values:
[tex]\[ x_1 = \frac{-(-1) + \sqrt{121}}{2 \cdot 3} \][/tex]
[tex]\[ x_1 = \frac{1 + 11}{6} \][/tex]
[tex]\[ x_1 = \frac{12}{6} \][/tex]
[tex]\[ x_1 = 2 \][/tex]
And for the second solution:
[tex]\[ x_2 = \frac{-(-1) - \sqrt{121}}{2 \cdot 3} \][/tex]
[tex]\[ x_2 = \frac{1 - 11}{6} \][/tex]
[tex]\[ x_2 = \frac{-10}{6} \][/tex]
[tex]\[ x_2 = -\frac{10}{6} \][/tex]
[tex]\[ x_2 = -\frac{5}{3} \][/tex]
[tex]\[ x_2 = -1.6666666666666667 \][/tex]
Thus, the solutions to the quadratic equation [tex]\(3x^2 - x - 10 = 0\)[/tex] are:
[tex]\[ x_1 = 2 \][/tex]
[tex]\[ x_2 = -1.6666666666666667 \][/tex]
The discriminant, [tex]\( \Delta \)[/tex], is 121.
So the solution for the value [tex]\( b \)[/tex] is:
[tex]\[ b = -1 \][/tex]
