Questions: Use the quadratic formula to find the real solutions, if any, of the equation. [ x^2+2 x-13=0 ]

Solution Steps

To solve the quadratic equation \(x^2 + 2x - 13 = 0\) using the quadratic formula, we need to identify the coefficients \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\). Then, we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the real solutions.

Given the quadratic equation \(x^2 + 2x - 13 = 0\), we identify the coefficients: \[ a = 1, \quad b = 2, \quad c = -13 \]

The discriminant \(\Delta\) of the quadratic equation is given by: \[ \Delta = b^2 - 4ac \] Substituting the values of \(a\), \(b\), and \(c\): \[ \Delta = 2^2 - 4 \cdot 1 \cdot (-13) = 4 + 52 = 56 \]

The quadratic formula to find the roots of the equation \(ax^2 + bx + c = 0\) is: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values of \(a\), \(b\), and \(\Delta\): \[ x = \frac{-2 \pm \sqrt{56}}{2 \cdot 1} = \frac{-2 \pm 7.4833}{2} \]

Calculating the two possible values for \(x\): \[ x_1 = \frac{-2 + 7.4833}{2} = \frac{5.4833}{2} = 2.7417 \] \[ x_2 = \frac{-2 - 7.4833}{2} = \frac{-9.4833}{2} = -4.7417 \]

Final Answer