2x²-5x-3=0

To solve the quadratic equation \(2x^2 - 5x - 3 = 0\), you can use the quadratic formula:

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

where \(a = 2\), \(b = -5\), and \(c = -3\).

1. Calculate the discriminant (\(b^2 - 4ac\)):

\[ b^2 - 4ac = (-5)^2 - 4(2)(-3) = 25 + 24 = 49 \]

2. Substitute the values into the quadratic formula:

\[ x = \frac{-(-5) \pm \sqrt{49}}{2(2)} = \frac{5 \pm 7}{4} \]

3. This gives us two potential solutions:

• For the positive case:

\[ x = \frac{5 + 7}{4} = \frac{12}{4} = 3 \]

• For the negative case:

\[ x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2} \]

So, the solutions to the equation \(2x^2 - 5x - 3 = 0\) are:

\[ x = 3 \quad \text{and} \quad x = -\frac{1}{2} \]