Here you'll learn more about quadratic equations and quadratic functions. You will learn three new methods for solving quadratic equations and discover the connections between a quadratic equation and its corresponding quadratic function. You will discover a new set of numbers called complex numbers and see how complex numbers are related to quadratic functions with no x-intercepts. Finally, you will learn how to solve radical equations.
You learned that all quadratic equations have a corresponding quadratic function. Real solutions to quadratic equations are the x-intercepts of the quadratic function. If a quadratic equation has only complex solutions, the quadratic function will not have x-intercepts.
You also learned that there are four methods for solving quadratic equations:
- Factoring and the zero product property (learned previously)
- Graphing and looking for x-intercepts
- Completing the square
- The quadratic formula: x=−b±b2−4ac−−−−−−−√2a
The advantage of the quadratic formula is that it will always work to give you solutions, even if the solutions are not real numbers.
If you want to determine whether the roots of a given quadratic function are real or complex, but you don't need to know specifically what the roots are, you can use the discriminant. The discriminant is the part of the quadratic formula under the square root symbol (b2−4ac). If the discriminant is negative, the roots will be complex. If the discriminant is equal to zero, there will only be one root (of multiplicity 2). If the discriminant is positive, the roots will be real.
You also learned that radical equations are equations with variables under square roots. Radical equations can be solved by isolating the square root and squaring both sides. Sometimes radical equations will produce extraneous solutions, which are not really solutions, so it is important to always check your answers to radical equations.