Questions: Use the quadratic formula to find the real solutions, if any, of the equation. x^2 + 2x - 22 = 0 Select the correct choice below and fill in any answer boxes within your choice. A. The solution set is (Simplify your answer. Type an exact answer, using radicals as needed. Use a B. There are no real solutions.

Solution Steps

To solve the quadratic equation \(x^2 + 2x - 22 = 0\), 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 from the quadratic equation \(ax^2 + bx + c = 0\). In this case, \(a = 1\), \(b = 2\), and \(c = -22\). We will calculate the discriminant \(b^2 - 4ac\) to determine if there are real solutions. If the discriminant is non-negative, we will compute the solutions using the quadratic formula.

The given quadratic equation is \(x^2 + 2x - 22 = 0\). From this equation, we identify the coefficients as follows:

• \(a = 1\)
• \(b = 2\)
• \(c = -22\)

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 \(c\), we get: \[ \Delta = 2^2 - 4 \times 1 \times (-22) = 4 + 88 = 92 \]

Since the discriminant \(\Delta = 92\) is positive, the quadratic equation has two distinct real solutions.

The solutions of the quadratic equation are given by the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values of \(b\), \(\Delta\), and \(a\), we find: \[ x_1 = \frac{-2 + \sqrt{92}}{2 \times 1} = \frac{-2 + 9.5917}{2} = \frac{7.5917}{2} = 3.7959 \] \[ x_2 = \frac{-2 - \sqrt{92}}{2 \times 1} = \frac{-2 - 9.5917}{2} = \frac{-11.5917}{2} = -5.7959 \]

Final Answer