To solve the quadratic equation \(10x^2 - 2x + 13 = 3\), we first move all terms to one side of the equation to set it to zero:
\[ 10x^2 - 2x + 13 - 3 = 0 \] This simplifies to:
\[ 10x^2 - 2x + 10 = 0 \]
Now, we can apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \(a = 10\), \(b = -2\), and \(c = 10\). Plugging in the values, we first calculate \(b^2 - 4ac\):
\[ b^2 = (-2)^2 = 4 \] \[ 4ac = 4 \cdot 10 \cdot 10 = 400 \] \[ b^2 - 4ac = 4 - 400 = -396 \]
Since the discriminant is negative (\(-396\)), we will have complex solutions. Now substituting back into the quadratic formula:
\[ x = \frac{-(-2) \pm \sqrt{-396}}{2 \cdot 10} \] \[ x = \frac{2 \pm \sqrt{-396}}{20} \]
To express \(\sqrt{-396}\):
\[ \sqrt{-396} = \sqrt{396}i = \sqrt{4 \cdot 99}i = 2\sqrt{99}i = 2\sqrt{9 \cdot 11}i = 6\sqrt{11}i \]
Now substituting this back into the equation:
\[ x = \frac{2 \pm 6\sqrt{11}i}{20} = \frac{1 \pm 3\sqrt{11}i}{10} \]
Thus, the solutions can be expressed as:
\[ x = \frac{1}{10} \pm \frac{3\sqrt{11}}{10}i \]
Now, looking at the responses you provided, the correct one is:
Positive \(\frac{1}{10} \pm \frac{3\sqrt{11}}{10} i\).
This matches the format: \[ \frac{1}{10} \pm \frac{3\sqrt{11}}{10}i \]
So the response is:
Start Fraction 1 over 10 End Fraction plus or minus Start Fraction 3 Start Root 11 End Root over 10 End Fraction i.
1 answer