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