Quadratic Equations Quick Check

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

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

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

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

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

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 will find the two possible values for \( x \):

1. For \( 5 + 7 \): \[ x = \frac{12}{4} = 3 \]

2. For \( 5 - 7 \): \[ x = \frac{-2}{4} = -\frac{1}{2} \]

Thus, the solutions are:

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

So the correct response is:

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