Solve the equation 0=2x2−5x−3

Question

algebraically using any method.(1 point)
Responses

x=−12
, x=3
x equals negative Start Fraction 1 over 2 End Fraction , x equals 3

x=−1
, x=32
x equals negative 1 , x equals Start Fraction 3 over 2 End Fraction

x=−3
, x=12
x equals negative 3 , x equals Start Fraction 1 over 2 End Fraction

x=−32
, x=1

Answer 1

To solve the equation \(0 = 2x^2 - 5x - 3\) algebraically, we will use the quadratic formula, which is given by:

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

In your equation, the coefficients are:

• \(a = 2\)
• \(b = -5\)
• \(c = -3\)

1. First, we need to calculate the discriminant \(b^2 - 4ac\):

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

2. Now we can substitute \(a\), \(b\), and the discriminant into the quadratic formula:

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

This simplifies to:

\[ x = \frac{5 \pm 7}{4} \]

3. Now we will calculate the two potential solutions:

Solution 1:

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

Solution 2:

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

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

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

Therefore, the correct response is:

x equals negative Start Fraction 1 over 2 End Fraction , x equals 3.