Use the quadratic formula to solve: 2x^2=x-5

The given quadratic equation is:

\[
2x^2 = x - 5
\]

First, rearrange it into standard quadratic form \( ax^2 + bx + c = 0 \). Subtract \( x - 5 \) from both sides:

\[
2x^2 - x + 5 = 0
\]

Now, use the **quadratic formula**:

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

For the equation \( 2x^2 - x + 5 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -1 \)
- \( c = 5 \)

Substitute these values into the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(5)}}{2(2)}
\]
\[
x = \frac{1 \pm \sqrt{1 - 40}}{4}
\]
\[
x = \frac{1 \pm \sqrt{-39}}{4}
\]

Since the discriminant \( \sqrt{-39} \) is negative, the solutions are complex numbers. Simplify:

\[
x = \frac{1 \pm i\sqrt{39}}{4}
\]

Thus, the two complex solutions are:

\[
x = \frac{1 + i\sqrt{39}}{4}, \quad x = \frac{1 - i\sqrt{39}}{4}
\]

Would you like more details on this solution or an explanation on working with complex numbers?

Here are 5 related questions to expand your understanding:
1. How do you determine the nature of the roots of a quadratic equation?
2. What happens when the discriminant is zero?
3. Can complex solutions appear in real-world problems?
4. How do you simplify expressions with imaginary numbers?
5. What are the steps to solve a quadratic equation by completing the square?

**Tip:** The discriminant \( b^2 - 4ac \) tells you if the solutions are real or complex: positive for real, zero for one real root, and negative for complex roots.

Use the quadratic formula to solve: 2x^2 = x - 5

Learn how to solve the quadratic equation 2x² = x - 5 using the quadratic formula. This solution involves working with complex numbers, utilizing the discriminant, and applying key algebraic concepts. Suitable for students in Grades 9-12.

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