Questions: Follow the Step-by-Step process to solve the equation by using the quadratic formula. x^2 + 10x = 14 x = [] (Simplify your answer. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals and i as needed.) Get more help - Clear all Check answer

Solution Steps

We start with the equation: \[ x^2 + 10x = 14 \] To rewrite it in standard form, we subtract 14 from both sides: \[ x^2 + 10x - 14 = 0 \]

In the standard form \(ax^2 + bx + c = 0\), we identify the coefficients:

• \(a = 1\)
• \(b = 10\)
• \(c = -14\)

We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting the values of \(a\), \(b\), and \(c\): \[ x = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot (-14)}}{2 \cdot 1} \]

Calculating the discriminant: \[ b^2 - 4ac = 100 + 56 = 156 \] Thus, the expression becomes: \[ x = \frac{-10 \pm \sqrt{156}}{2} \] We can simplify \(\sqrt{156}\): \[ \sqrt{156} = \sqrt{4 \cdot 39} = 2\sqrt{39} \] Now substituting back: \[ x = \frac{-10 \pm 2\sqrt{39}}{2} \] This simplifies to: \[ x = -5 \pm \sqrt{39} \]

The solutions to the equation \(x^2 + 10x = 14\) are: \[ x = -5 + \sqrt{39}, \quad x = -5 - \sqrt{39} \]

Final Answer