Questions: 6x^2 + 4x - 3 = 0

Solution Steps

The given quadratic equation is

\[ 6x^{2} + 4x - 3 = 0 \]

From this equation, we identify the coefficients as follows:

• \( a = 6 \)
• \( b = 4 \)
• \( c = -3 \)

The discriminant \( D \) is calculated using the formula:

\[ D = b^{2} - 4ac \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = 4^{2} - 4 \cdot 6 \cdot (-3) = 16 + 72 = 88 \]

The roots of the quadratic equation can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

Substituting the values of \( b \), \( D \), and \( a \):

\[ x = \frac{-4 \pm \sqrt{88}}{2 \cdot 6} \]

Calculating the square root:

\[ \sqrt{88} = \sqrt{4 \cdot 22} = 2\sqrt{22} \]

Thus, the roots become:

\[ x = \frac{-4 \pm 2\sqrt{22}}{12} = \frac{-2 \pm \sqrt{22}}{6} \]

The two roots can be expressed as:

\[ x_1 = \frac{-2 + \sqrt{22}}{6}, \quad x_2 = \frac{-2 - \sqrt{22}}{6} \]

Calculating the approximate values:

\[ x_1 \approx 0.4484, \quad x_2 \approx -1.1151 \]

Final Answer