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