Questions: Solve the equation: (9 t^2+6 t-79=0). Fully simplify your answer, including any non-real solutions. (t=)

Solution Steps

To solve the quadratic equation \(9t^2 + 6t - 79 = 0\), we can use the quadratic formula, which is given by \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 9\), \(b = 6\), and \(c = -79\). We will calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots and then apply the quadratic formula to find the solutions.

The given quadratic equation is \(9t^2 + 6t - 79 = 0\). We identify the coefficients as follows:

• \(a = 9\)
• \(b = 6\)
• \(c = -79\)

The discriminant \(\Delta\) of a quadratic equation \(at^2 + bt + c = 0\) is given by: \[ \Delta = b^2 - 4ac \] Substituting the values, we have: \[ \Delta = 6^2 - 4 \times 9 \times (-79) = 36 + 2844 = 2880 \]

The solutions for the quadratic equation are given by the quadratic formula: \[ t = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values, we calculate: \[ t_1 = \frac{-6 + \sqrt{2880}}{18} \approx 2.648 \] \[ t_2 = \frac{-6 - \sqrt{2880}}{18} \approx -3.315 \]

Final Answer