16 Multiple Choice 5 points Solve the following equation. 2x^2+13=-10x x=(-5pm i)/(2) x=10pm 4 x=5pm 2i No solution

To solve the equation \(2x^2 + 13 = -10x\), we first rearrange it into standard quadratic form:<br /><br />\[ <br />2x^2 + 10x + 13 = 0 <br />\]<br /><br />This is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 2\), \(b = 10\), and \(c = 13\).<br /><br />We can use the quadratic formula to find the solutions for \(x\):<br /><br />\[<br />x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}<br />\]<br /><br />Substitute the values of \(a\), \(b\), and \(c\) into the formula:<br /><br />\[<br />x = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 2 \cdot 13}}{2 \cdot 2}<br />\]<br /><br />Calculate the discriminant:<br /><br />\[<br />b^2 - 4ac = 10^2 - 4 \cdot 2 \cdot 13 = 100 - 104 = -4<br />\]<br /><br />Since the discriminant is negative, the solutions will be complex numbers. Calculate the square root of the discriminant:<br /><br />\[<br />\sqrt{-4} = 2i<br />\]<br /><br />Now substitute back into the quadratic formula:<br /><br />\[<br />x = \frac{-10 \pm 2i}{4}<br />\]<br /><br />Simplify the expression:<br /><br />\[<br />x = \frac{-10}{4} \pm \frac{2i}{4}<br />\]<br />\[<br />x = -\frac{5}{2} \pm \frac{i}{2}<br />\]<br /><br />Thus, the solutions are:<br /><br />\[<br />x = \frac{-5 \pm i}{2}<br />\]<br /><br />Therefore, the correct answer is:<br /><br />\( x=\frac{-5 \pm i}{2} \)