Questions: Solve the equation. [ x(5 x+12)=9 ] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is ∅.

Solution Steps

To solve the equation \( x(5x + 12) = 9 \), we first expand the left side to form a quadratic equation. Then, we rearrange the equation to standard quadratic form \( ax^2 + bx + c = 0 \). Next, we use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solutions for \( x \). Finally, we check if the solutions are real numbers or if the solution set is empty.

Starting with the equation \( x(5x + 12) = 9 \), we expand it to obtain: \[ 5x^2 + 12x - 9 = 0 \] Here, we identify the coefficients: \( a = 5 \), \( b = 12 \), and \( c = -9 \).

We calculate the discriminant \( D \) using the formula: \[ D = b^2 - 4ac \] Substituting the values, we find: \[ D = 12^2 - 4 \cdot 5 \cdot (-9) = 144 + 180 = 324 \]

Since the discriminant is positive (\( D = 324 \)), we can find two real solutions using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Calculating the two roots: \[ x_1 = \frac{-12 + \sqrt{324}}{2 \cdot 5} = \frac{-12 + 18}{10} = \frac{6}{10} = \frac{3}{5} \] \[ x_2 = \frac{-12 - \sqrt{324}}{2 \cdot 5} = \frac{-12 - 18}{10} = \frac{-30}{10} = -3 \]

Final Answer