To solve the quadratic equation \( 0 = 2x^2 - 5x - 3 \) algebraically, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation, the coefficients are:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -3 \)
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 2 \cdot (-3) = 25 + 24 = 49 \]
Now, we can plug the values into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{49}}{2 \cdot 2} \] \[ x = \frac{5 \pm 7}{4} \]
Now, we solve for the two possible values of \( x \):
- \( x = \frac{5 + 7}{4} = \frac{12}{4} = 3 \)
- \( x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2} \)
Thus, the solutions to the equation \( 0 = 2x^2 - 5x - 3 \) are:
\[ x = 3 \quad \text{and} \quad x = -\frac{1}{2} \]
Among the provided response options, the correct solution is:
x equals negative Start Fraction 1 over 2 End Fraction , x equals 3.