Questions: 2x^2 + 3x - 27 = 0

Solution Steps

To solve the quadratic equation \(2x^2 + 3x - 27 = 0\), we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the equation \(ax^2 + bx + c = 0\).

Given the quadratic equation: \[ 2x^2 + 3x - 27 = 0 \] The coefficients are: \[ a = 2, \quad b = 3, \quad c = -27 \]

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by: \[ \Delta = b^2 - 4ac \] Substituting the values of \(a\), \(b\), and \(c\): \[ \Delta = 3^2 - 4 \cdot 2 \cdot (-27) = 9 + 216 = 225 \]

The solutions to the quadratic equation are given by: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values of \(a\), \(b\), and \(\Delta\): \[ x = \frac{-3 \pm \sqrt{225}}{2 \cdot 2} = \frac{-3 \pm 15}{4} \]

Solving for the two possible values of \(x\): \[ x_1 = \frac{-3 + 15}{4} = \frac{12}{4} = 3 \] \[ x_2 = \frac{-3 - 15}{4} = \frac{-18}{4} = -4.5 \]

Final Answer