Questions: Use the quadratic formula to solve 3x^2-x+10=0 x=(-b ± sqrt(b^2-4ac))/(2a) (1+sqrt(119)i)/3 and (1-sqrt(119)i)/3 (1+sqrt(119)i)/6 and (1-sqrt(119)i)/6 (-1+sqrt(119)i)/3 and (-1-sqrt(119)i)/3 (-1+sqrt(119)i)/6 and (-1-sqrt(119)i)/6

Use the quadratic formula to solve 3x^2-x+10=0 x=(-b ± sqrt(b^2-4ac))/(2a) (1+sqrt(119)i)/3 and (1-sqrt(119)i)/3 (1+sqrt(119)i)/6 and (1-sqrt(119)i)/6 (-1+sqrt(119)i)/3 and (-1-sqrt(119)i)/3 (-1+sqrt(119)i)/6 and (-1-sqrt(119)i)/6
Transcript text: Use the quadratic formula to solve $3 x^{2}-x+10=0$ \[ x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \] $\frac{1+\sqrt{119} i}{3}$ and $\frac{1-\sqrt{119} i}{3}$ $\frac{1+\sqrt{119} i}{6}$ and $\frac{1-\sqrt{119} i}{6}$ $\frac{-1+\sqrt{119} i}{3}$ and $\frac{-1-\sqrt{119} i}{3}$ $\frac{-1+\sqrt{119} i}{6}$ and $\frac{-1-\sqrt{119} i}{6}$
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Solution

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Solution Steps

To solve the quadratic equation \(3x^2 - x + 10 = 0\) using the quadratic formula, identify the coefficients \(a = 3\), \(b = -1\), and \(c = 10\). Substitute these values into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots of the equation. Since the discriminant \(b^2 - 4ac\) is negative, the solutions will be complex numbers.

Step 1: Identify Coefficients

For the quadratic equation \(3x^2 - x + 10 = 0\), we identify the coefficients as follows:

  • \(a = 3\)
  • \(b = -1\)
  • \(c = 10\)
Step 2: Calculate the Discriminant

The discriminant \(D\) is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values, we have: \[ D = (-1)^2 - 4 \cdot 3 \cdot 10 = 1 - 120 = -119 \]

Step 3: Apply the Quadratic Formula

Since the discriminant is negative, the solutions will be complex. We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting the values, we find: \[ x = \frac{1 \pm \sqrt{-119}}{6} \] This simplifies to: \[ x = \frac{1 \pm i\sqrt{119}}{6} \]

Final Answer

The solutions to the equation are: \[ x_1 = \frac{1 + i\sqrt{119}}{6}, \quad x_2 = \frac{1 - i\sqrt{119}}{6} \] Thus, the final boxed answers are: \[ \boxed{x_1 = \frac{1 + i\sqrt{119}}{6}, \quad x_2 = \frac{1 - i\sqrt{119}}{6}} \]

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