To solve the quadratic equation [tex]\(3x^2 - x = 10\)[/tex], we will first rewrite the equation in standard form:
[tex]\[3x^2 - x - 10 = 0\][/tex]
This is a standard quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 3\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -10\)[/tex].
To solve this quadratic equation, we use the quadratic formula:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 3 \cdot (-10)}}{2 \cdot 3}\][/tex]
Simplify inside the square root:
[tex]\[x = \frac{1 \pm \sqrt{1 + 120}}{6}\][/tex]
[tex]\[x = \frac{1 \pm \sqrt{121}}{6}\][/tex]
Since [tex]\(\sqrt{121} = 11\)[/tex], we have:
[tex]\[x = \frac{1 \pm 11}{6}\][/tex]
This gives us two possible solutions:
1. [tex]\(x = \frac{1 + 11}{6} = \frac{12}{6} = 2\)[/tex]
2. [tex]\(x = \frac{1 - 11}{6} = \frac{-10}{6} = -\frac{5}{3} \)[/tex]
Therefore, the solutions to the equation [tex]\(3x^2 - x = 10\)[/tex] are:
[tex]\[x = 2 \quad \text{and} \quad x = -\frac{5}{3}\][/tex]
Among the given options, the correct answer is:
[tex]\[x = -\frac{5}{3} \text{ and } x = 2\][/tex]
