Answer:
[tex]x_1 = \frac{3}{2} + \frac{1}{2}(\sqrt{47})i\\\\x_2 = \frac{3}{2} - \frac{1}{2}(\sqrt{47})i\\\\[/tex]
Step-by-step explanation:
In this problem we have the equation of the following quadratic equation and we want to solve it using the method of square completion:
[tex]x ^ 2 -3x +14 = 0[/tex]
The steps are shown below:
For any equation of the form: [tex]ax ^ 2 + bx + c = 0[/tex]
1. If the coefficient a is different from 1, then take a as a common factor.
In this case [tex]a = 1[/tex].
Then we go directly to step 2
2. Take the coefficient b that accompanies the variable x. In this case the coefficient is -3. Then, divide by 2 and the result squared it.
We have:
[tex]\frac{-3}{2} = -\frac{3}{2}\\\\(-\frac{3}{2}) ^ 2 = (\frac{9}{4})[/tex]
3. Add the term obtained in the previous step on both sides of equality:
[tex]x ^ 2 -3x + (\frac{9}{4}) = -14 + (\frac{9}{4})[/tex]
4. Factor the resulting expression, and you will get:
[tex](x -\frac{3}{2}) ^ 2 = -(\frac{47}{4})[/tex]
Now solve the equation:
Note that the term [tex](x -\frac{3}{2}) ^ 2[/tex] is always > 0 therefore it can not be equal to [tex]-(\frac{47}{4})[/tex]
The equation has no solution in real numbers.
In the same way we can find the complex roots:
[tex](x -\frac{3}{2}) ^ 2 = -(\frac{47}{4})\\\\x -\frac{3}{2} = \±\sqrt{-(\frac{47}{4})}\\\\x = \frac{3}{2} \±\frac{1}{2}\sqrt{-47}\\\\x = \frac{3}{2} \±\frac{1}{2}(\sqrt{47})i\\\\x_1 = \frac{3}{2} + \frac{1}{2}(\sqrt{47})i\\\\x_2 = \frac{3}{2} - \frac{1}{2}(\sqrt{47})i\\\\[/tex]
