3y²-2y+7=0 solve for y by splitting of middle term

To solve the quadratic equation \( 3y^2 - 2y + 7 = 0 \) by splitting the middle term, we follow these steps:

1. **Identify the coefficients**:  
   The equation is in the form \( ay^2 + by + c = 0 \), where:
   - \( a = 3 \)
   - \( b = -2 \)
   - \( c = 7 \)

2. **Multiply \( a \) and \( c \)**:  
   Multiply the coefficient of \( y^2 \) (which is \( a = 3 \)) by the constant term \( c = 7 \):
   \[
   3 \times 7 = 21
   \]

3. **Find two numbers that multiply to \( ac \) and add to \( b \)**:  
   We need to find two numbers that multiply to \( 21 \) and add to \( -2 \). However, no real numbers satisfy these conditions because the product is positive and the sum is negative. 

   This indicates that the middle term cannot be split in such a way that we get rational factors, and we must resort to other methods like completing the square or using the quadratic formula.

4. **Use the quadratic formula** (since splitting isn't feasible):
   The quadratic formula is:
   \[
   y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Plugging in the values:
   \[
   y = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(7)}}{2(3)}
   \]
   \[
   y = \frac{2 \pm \sqrt{4 - 84}}{6}
   \]
   \[
   y = \frac{2 \pm \sqrt{-80}}{6}
   \]
   \[
   y = \frac{2 \pm \sqrt{80}i}{6}
   \]
   \[
   y = \frac{2 \pm 4\sqrt{5}i}{6}
   \]
   \[
   y = \frac{1 \pm \frac{2\sqrt{5}i}{3}}{3}
   \]

So the solutions are:
\[
y = \frac{1}{3} + \frac{2\sqrt{5}i}{3}, \quad y = \frac{1}{3} - \frac{2\sqrt{5}i}{3}
\]

These solutions are complex numbers since the discriminant is negative.

Would you like more details on any step, or have any questions?

### Related Questions
1. How do you solve quadratic equations with real roots by completing the square?
2. What is the geometric interpretation of complex roots in a quadratic equation?
3. How do you determine the nature of roots without solving the quadratic equation?
4. Can all quadratic equations be solved by splitting the middle term?
5. How does the discriminant relate to the type of roots in a quadratic equation?

### Tip
When the discriminant (\(b^2 - 4ac\)) is negative, the quadratic equation has complex roots.

Learn how to solve the quadratic equation 3y² - 2y + 7 = 0 using the method of splitting the middle term. This problem involves complex numbers and covers key algebraic concepts. Suitable for students in Grades 10-12.

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