Certainly! Let's solve the quadratic equation [tex]\(3x^2 - 2x - 6 = 0\)[/tex] step by step and round the solutions to the nearest thousandth.
1. Identify the coefficients:
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex].
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -6 \)[/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].
Plugging in the values, we get:
[tex]\[
\Delta = (-2)^2 - 4 \cdot 3 \cdot (-6) = 4 + 72 = 76
\][/tex]
3. Calculate the two solutions using the quadratic formula:
The solutions of [tex]\( ax^2 + bx + c = 0 \)[/tex] are given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Plugging in the values, we have:
[tex]\[
x_1 = \frac{-(-2) + \sqrt{76}}{2 \cdot 3} = \frac{2 + \sqrt{76}}{6}
\][/tex]
[tex]\[
x_2 = \frac{-(-2) - \sqrt{76}}{2 \cdot 3} = \frac{2 - \sqrt{76}}{6}
\][/tex]
4. Simplify the solutions and round to the nearest thousandth:
Here are the calculations:
[tex]\[
x_1 = \frac{2 + \sqrt{76}}{6} \approx 1.786
\][/tex]
[tex]\[
x_2 = \frac{2 - \sqrt{76}}{6} \approx -1.120
\][/tex]
5. Final solutions rounded to the nearest thousandth:
The solutions to the quadratic equation [tex]\( 3x^2 - 2x - 6 = 0 \)[/tex] are approximately:
[tex]\[
x_1 \approx 1.786
\][/tex]
[tex]\[
x_2 \approx -1.120
\][/tex]
These are the solutions rounded to the nearest thousandth.
