Question

Find the roots of the quadratic equation \(4x^2 + 9x + 5 = 0.\)

Hint:

Always check discriminant \(D = b^2 - 4ac\) to know the nature of roots before solving.

Answer 1

Step 1: Identify coefficients.
\(a = 4, \, b = 9, \, c = 5\).
Step 2: Apply quadratic formula.
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Step 3: Substitute values.
\[ x = \frac{-9 \pm \sqrt{9^2 - 4(4)(5)}}{8} = \frac{-9 \pm \sqrt{81 - 80}}{8} = \frac{-9 \pm 1}{8} \]
Step 4: Calculate roots.
\[ x_1 = \frac{-9 + 1}{8} = -1, \quad x_2 = \frac{-9 - 1}{8} = -\frac{5}{2} \]
Step 5: Conclusion.
Hence, the roots are \(\boxed{x = -1 \text{ and } x = -\frac{5}{2}}\).