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 3x2−x+10=03x^2 - x + 10 = 0 using the quadratic formula, identify the coefficients a=3a = 3, b=−1b = -1, and c=10c = 10. Substitute these values into the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} to find the roots of the equation. Since the discriminant b2−4acb^2 - 4ac is negative, the solutions will be complex numbers.

Step 1: Identify Coefficients

For the quadratic equation 3x2−x+10=03x^2 - x + 10 = 0, we identify the coefficients as follows:

  • a=3a = 3
  • b=−1b = -1
  • c=10c = 10
Step 2: Calculate the Discriminant

The discriminant DD is calculated using the formula: D=b2−4ac D = b^2 - 4ac Substituting the values, we have: D=(−1)2−4⋅3⋅10=1−120=−119 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=−b±D2a x = \frac{-b \pm \sqrt{D}}{2a} Substituting the values, we find: x=1±−1196 x = \frac{1 \pm \sqrt{-119}}{6} This simplifies to: x=1±i1196 x = \frac{1 \pm i\sqrt{119}}{6}

Final Answer

The solutions to the equation are: x1=1+i1196,x2=1−i1196 x_1 = \frac{1 + i\sqrt{119}}{6}, \quad x_2 = \frac{1 - i\sqrt{119}}{6} Thus, the final boxed answers are: x1=1+i1196,x2=1−i1196 \boxed{x_1 = \frac{1 + i\sqrt{119}}{6}, \quad x_2 = \frac{1 - i\sqrt{119}}{6}}

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