Questions: Use the quadratic formula to solve for (x). [ 7 x^2+3 x-2=0 ] (If there is more than one solution, separate them with commas.) [ x= ]

Solution Steps

To solve the quadratic equation \(7x^2 + 3x - 2 = 0\), we will use the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\). In this case, \(a = 7\), \(b = 3\), and \(c = -2\). We will calculate the discriminant \(b^2 - 4ac\) to determine the solutions.

The given quadratic equation is \(7x^2 + 3x - 2 = 0\). The coefficients are:

• \(a = 7\)
• \(b = 3\)
• \(c = -2\)

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by: \[ \Delta = b^2 - 4ac \] Substituting the values, we have: \[ \Delta = 3^2 - 4 \times 7 \times (-2) = 9 + 56 = 65 \]

The solutions for \(x\) are given by the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values, we find: \[ x_1 = \frac{-3 + \sqrt{65}}{14} \] \[ x_2 = \frac{-3 - \sqrt{65}}{14} \]

Calculating the numerical values:

• \(x_1 \approx 0.3616\)
• \(x_2 \approx -0.7902\)

Final Answer