Question

Solve the quadratic equation.
4x^2+5x+2=0

Answer 1

To solve the quadratic equation \( 4x^2 + 5x + 2 = 0 \), we can use the quadratic formula:

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

In this equation, the coefficients are:
- \( a = 4 \)
- \( b = 5 \)
- \( c = 2 \)

Now, we will calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = 5^2 - 4 \cdot 4 \cdot 2 = 25 - 32 = -7
\]

Since the discriminant is negative (\( -7 \)), the solutions will be complex numbers. Now we can substitute into the quadratic formula:

\[
x = \frac{-5 \pm \sqrt{-7}}{2 \cdot 4}
\]

This simplifies to:

\[
x = \frac{-5 \pm i\sqrt{7}}{8}
\]

Thus, the solutions to the quadratic equation \( 4x^2 + 5x + 2 = 0 \) are:

\[
x = \frac{-5 + i\sqrt{7}}{8} \quad \text{and} \quad x = \frac{-5 - i\sqrt{7}}{8}
\]