Questions: Solve the following quadratic equation by using the quadratic formula. 6x^2 + 6x = 7 x= (Simplify your answer. Type an exact answer, using radicals and i as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

Solution Steps

The given quadratic equation is

\[ 6x^{2} + 6x - 7 = 0 \]

The coefficients are \( a = 6 \), \( b = 6 \), and \( c = -7 \). The discriminant \( D \) is calculated as follows:

\[ D = b^2 - 4ac = 6^2 - 4 \cdot 6 \cdot (-7) = 36 + 168 = 204 \]

Since the discriminant is positive, we can find two real solutions using the quadratic formula:

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

Substituting the values:

\[ x = \frac{-6 \pm \sqrt{204}}{2 \cdot 6} \]

The square root of 204 can be simplified:

\[ \sqrt{204} = \sqrt{4 \cdot 51} = 2\sqrt{51} \]

Now substituting back into the formula gives:

\[ x = \frac{-6 \pm 2\sqrt{51}}{12} \]

This simplifies to:

\[ x = \frac{-1 \pm \frac{1}{6}\sqrt{51}}{2} \]

Thus, the two solutions are:

\[ x_1 = \frac{-1 + \sqrt{51}}{6}, \quad x_2 = \frac{-1 - \sqrt{51}}{6} \]

Final Answer