what are the roots of the equation $x^{2}+6x + 10 = 0$ in simplest $a+bi$ form?
what are the roots of the equation $x^{2}+6x + 10 = 0$ in simplest $a+bi$ form?
Answer
Explanation:
Step1: Identify coefficients
For the quadratic equation (x^{2}+6x + 10=0), (a = 1), (b=6), (c = 10).
Step2: Use quadratic formula
The quadratic formula is (x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}). Substitute (a = 1), (b = 6), (c=10) into the formula: [ \begin{align*} x&=\frac{-6\pm\sqrt{6^{2}-4\times1\times10}}{2\times1}\ &=\frac{-6\pm\sqrt{36 - 40}}{2}\ &=\frac{-6\pm\sqrt{-4}}{2} \end{align*} ] Since (\sqrt{-4}=\sqrt{4\times(- 1)} = 2i), then (x=\frac{-6\pm2i}{2}).
Step3: Simplify the expression
Divide each term in the numerator by (2): (x=-3\pm i)
Answer:
(x=-3 + i) and (x=-3 - i)