Learning Outcomes
- Use the quadratic formula to solve quadratic equations with complex solutions
We have seen two outcomes for solutions to quadratic equations; either there was one or two real number solutions. We have also learned that it is possible to take the square root of a negative number by using imaginary numbers. Having this new knowledge allows us to explore one more possible outcome when we solve quadratic equations. Consider this equation:
[latex]2x^2+3x+6=0[/latex]
Using the Quadratic Formula to solve this equation, we first identify a, b, and c.
[latex]a = 2,b = 3,c = 6[/latex]
We can place a, b and c into the quadratic formula and simplify to get the following result:
[latex]x=-\frac{3}{4}+\frac{\sqrt{-39}}{4}, x=-\frac{3}{4}-\frac{\sqrt{-39}}{4}[/latex]
Up to this point, we would have said that [latex]\sqrt{-39}[/latex] is not defined for real numbers and determine that this equation has no solutions. But, now that we have defined the square root of a negative number, we can also define a solution to this equation as follows.
[latex]x=-\frac{3}{4}+i\frac{\sqrt{39}}{4}, x=-\frac{3}{4}-i\frac{\sqrt{39}}{4}[/latex]
In this section we will practice simplifying and writing solutions to quadratic equations that are complex. We will then present a technique for classifying whether the solution(s) to a quadratic equation will be complex, and how many solutions there will be.
In our first example, we will work through the process of solving a quadratic equation with complex solutions. Take note that we will be simplifying complex numbers, so if you need a review of how to rewrite the square root of a negative number as an imaginary number, now is a good time.
Example
Use the Quadratic Formula to solve the equation [latex]x^{2}+2x=-5[/latex].
We can check these solutions in the original equation. Be careful when you expand the squares, and replace [latex]i^{2}[/latex] with [latex]-1[/latex].
| [latex]\begin{array}{r}x=-1+2i\\x^{2}+2x=-5\\\left(-1+2i\right)^{2}+2\left(-1+2i\right)=-5\\1-4i+4i^{2}-2+4i=-5\\1-4i+4\left(-1\right)-2+4i=-5\\1-4-2=-5\\-5=-5\end{array}[/latex] | [latex]\begin{array}{r}x=-1-2i\\x^{2}+2x=-5\\\left(-1-2i\right)^{2}+2\left(-1-2i\right)=-5\\1+4i+4i^{2}-2-4i=-5\\1+4i+4\left(-1\right)-2-4i=-5\\1-4-2=-5\\-5=-5\end{array}[/latex] |
Example
Use the quadratic formula to solve [latex]{x}^{2}+x+2=0[/latex].
Try It
The following video gives another example of how to use the quadratic formula to find solutions to a quadratic equation that has complex solutions.
Summary
Quadratic equations can have complex solutions.