To solve the equation \(0 = 2x^2 - 5x - 3\) algebraically, we can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
Here, we have:
- \(a = 2\)
- \(b = -5\)
- \(c = -3\)
First, we calculate the discriminant \(b^2 - 4ac\):
\[ b^2 - 4ac = (-5)^2 - 4(2)(-3) = 25 + 24 = 49 \]
Next, we apply the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{49}}{2 \times 2} = \frac{5 \pm 7}{4} \]
Now we calculate the two possible values for \(x\):
- \(x = \frac{5 + 7}{4} = \frac{12}{4} = 3\)
- \(x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}\)
So the solutions to the equation \(2x^2 - 5x - 3 = 0\) are:
\[ x = 3 \quad \text{and} \quad x = -\frac{1}{2} \]
Thus, the final answer is:
Responses:
- \(x = 3\), \(x = -\frac{1}{2}\)
1 answer