To solve the quadratic equation \( 5x^2 = -4x + 2 \), follow these steps:

1. **Rewrite the equation**: First, rearrange the equation into standard quadratic form, \( ax^2 + bx + c = 0 \).
   \[
   5x^2 + 4x - 2 = 0
   \]

2. **Apply the quadratic formula**: The quadratic formula is given by:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 5 \), \( b = 4 \), and \( c = -2 \).

3. **Substitute the values**:
   \[
   x = \frac{-4 \pm \sqrt{4^2 - 4(5)(-2)}}{2(5)}
   \]
   Simplify under the square root:
   \[
   x = \frac{-4 \pm \sqrt{16 + 40}}{10}
   \]
   \[
   x = \frac{-4 \pm \sqrt{56}}{10}
   \]

4. **Simplify the square root**:
   \[
   \sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14}
   \]
   So, the equation becomes:
   \[
   x = \frac{-4 \pm 2\sqrt{14}}{10}
   \]

5. **Simplify further**:
   \[
   x = \frac{-2 \pm \sqrt{14}}{5}
   \]

Thus, the solutions are:
\[
x = \frac{-2 + \sqrt{14}}{5} \quad \text{or} \quad x = \frac{-2 - \sqrt{14}}{5}
\]

Would you like further details on any of these steps?

Here are some related questions to explore:
1. What is the quadratic formula and how is it derived?
2. How do you simplify expressions with square roots?
3. How do you determine the nature of the roots in a quadratic equation?
4. Can the quadratic equation be solved using factoring?
5. What are the applications of solving quadratic equations in real life?

**Tip:** Always check if a quadratic equation can be factored before applying the quadratic formula, as factoring might be quicker.

Learn how to solve the quadratic equation 5x^2 = -4x + 2 using the quadratic formula. This step-by-step solution simplifies the expression and explores key concepts such as solving quadratic equations and simplifying square roots. Suitable for students in Grades 9-10.

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