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To solve the quadratic equation \( y = x^2 - 3x + 8 \) using the quadratic formula, we follow these steps:

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( y = x^2 - 3x + 8 \), we identify the coefficients:
- \( a = 1 \) (the coefficient of \( x^2 \))
- \( b = -3 \) (the coefficient of \( x \))
- \( c = 8 \) (the constant term)

### Step 1: Calculate the discriminant
The discriminant \( \Delta \) is the part under the square root in the quadratic formula:

\[
\Delta = b^2 - 4ac
\]

Substituting the values of \( a \), \( b \), and \( c \):

\[
\Delta = (-3)^2 - 4(1)(8) = 9 - 32 = -23
\]

### Step 2: Interpret the discriminant
Since the discriminant is negative (\( \Delta = -23 \)), this quadratic equation does not have real solutions but has complex solutions.

### Step 3: Apply the quadratic formula
Now, we substitute the values into the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{-23}}{2(1)} = \frac{3 \pm \sqrt{-23}}{2}
\]

Since \( \sqrt{-23} = i\sqrt{23} \) (where \( i \) is the imaginary unit), the solutions are:

\[
x = \frac{3 \pm i\sqrt{23}}{2}
\]

Thus, the two complex solutions are:

\[
x = \frac{3 + i\sqrt{23}}{2} \quad \text{and} \quad x = \frac{3 - i\sqrt{23}}{2}
\]

Would you like further details or have any questions about this solution?

Here are 5 related questions to expand your understanding:
1. How does the discriminant determine the nature of the roots in a quadratic equation?
2. What are the real-life applications of complex solutions in quadratic equations?
3. How do we graph quadratic equations with complex solutions?
4. Can you derive the quadratic formula from completing the square?
5. How do the values of \( a \), \( b \), and \( c \) affect the shape and position of a parabola?

**Tip:** When the discriminant is negative, the solutions involve complex numbers, which always appear in conjugate pairs!

Solve using the quadratic formula for y = x^2 - 3x + 8

Learn how to solve the quadratic equation y = x^2 - 3x + 8 with a step-by-step solution. This problem explores algebraic concepts such as the quadratic equation, discriminant, and complex solutions. Suitable for Grades 8-10.

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