Questions: Solve the equation by using the quadratic formula 3 k^2-k-3=0 Simplify answers. List multiple answers with a comma in bet

Solution Steps

To solve the quadratic equation $3k^2 - k - 3 = 0$ using the quadratic formula, we need to identify the coefficients $a$, $b$, and $c$ from the equation $ax^2 + bx + c = 0$. Then, we apply the quadratic formula $k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions for $k$.

1. Identify the coefficients $a$, $b$, and $c$ from the equation.
2. Use the quadratic formula to calculate the roots.
3. Simplify the roots if possible.

The given quadratic equation is $3k^2 - k - 3 = 0$. From this equation, we identify the coefficients as follows:

• $a = 3$
• $b = -1$
• $c = -3$

We calculate the discriminant using the formula $D = b^2 - 4ac$: $$D = (-1)^2 - 4 \cdot 3 \cdot (-3) = 1 + 36 = 37$$

Using the quadratic formula $k = \frac{-b \pm \sqrt{D}}{2a}$, we find the two solutions: $$k_1 = \frac{-(-1) + \sqrt{37}}{2 \cdot 3} = \frac{1 + \sqrt{37}}{6} \approx 1.1805$$ $$k_2 = \frac{-(-1) - \sqrt{37}}{2 \cdot 3} = \frac{1 - \sqrt{37}}{6} \approx -0.8471$$

Final Answer