To solve the quadratic equation \(2x^2 - 5x - 3 = 0\), we can apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation \(2x^2 - 5x - 3\), the coefficients are:
- \(a = 2\)
- \(b = -5\)
- \(c = -3\)
Now, substitute these values into the quadratic formula:
- Calculate the discriminant:
\[ b^2 - 4ac = (-5)^2 - 4(2)(-3) = 25 + 24 = 49 \]
- Now substitute back into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{49}}{2 \cdot 2} = \frac{5 \pm 7}{4} \]
- Now we compute the two possible values for \(x\):
- For the positive root:
\[ x = \frac{5 + 7}{4} = \frac{12}{4} = 3 \]
- For the negative root:
\[ x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2} \]
Thus, the solutions to the equation \(2x^2 - 5x - 3 = 0\) are:
\[ x = 3 \quad \text{and} \quad x = -\frac{1}{2} \]