- Solutions of a Quadratic Equation
- one.-
- 2.- In complex numbers
- How are the solutions of a quadratic equation found?
- Examples:
- References
A quadratic equation or quadratic equation can have zero, one or two real solutions, depending on the coefficients that appear in said equation.
If you work on complex numbers then you can say that every quadratic equation has two solutions.
To begin with, a quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are real numbers and x is a variable.
It is said that x1 is a solution of the previous quadratic equation if replacing x by x1 satisfies the equation, that is, if a (x1) ² + b (x1) + c = 0.
If, for example, we have the equation x²-4x + 4 = 0, then x1 = 2 is a solution since (2) ²-4 (2) + 4 = 4-8 + 4 = 0.
On the contrary, if we substitute x2 = 0 we obtain (0) ²-4 (0) + 4 = 4 and since 4 ≠ 0 then x2 = 0 is not a solution of the quadratic equation.
Solutions of a Quadratic Equation
The number of solutions of a quadratic equation can be separated into two cases which are:
one.-
When working with real numbers, quadratic equations can have:
-Zero solutions: that is, there is no real number that satisfies the quadratic equation. For example, the equation given the equation x² + 1 = 0, there is no such real number that satisfies said equation, since both x² is greater than or equal to zero and 1 is strictly greater than zero, so their sum will be greater strict than zero.
-A repeated solution: there is a single real value that satisfies the quadratic equation. For example, the only solution to the equation x²-4x + 4 = 0 is x1 = 2.
-Two different solutions: there are two values that satisfy the quadratic equation. For example, x² + x-2 = 0 has two different solutions which are x1 = 1 and x2 = -2.
2.- In complex numbers
When working with complex numbers, quadratic equations always have two solutions which are z1 and z2 where z2 is the conjugate of z1. They can also be classified into:
-Complexes: the solutions are of the form z = p ± qi, where p and q are real numbers. This case corresponds to the first case in the previous list.
-Pure Complexes: is when the real part of the solution is equal to zero, that is, the solution has the form z = ± qi, where q is a real number. This case corresponds to the first case in the previous list.
-Complexes with an imaginary part equal to zero: it is when the complex part of the solution is equal to zero, that is, the solution is a real number. This case corresponds to the last two cases in the previous list.
How are the solutions of a quadratic equation found?
To calculate the solutions of a quadratic equation, a formula known as "the resolvent" is used, which says that the solutions of an equation ax² + bx + c = 0 are given by the expression in the following image:
The quantity that appears within the square root is called the discriminant of the quadratic equation and is denoted by the letter "d".
The quadratic equation will have:
-Two real solutions if, and only if, d> 0.
-A real solution repeated if, and only if, d = 0.
-Zero real solutions (or two complex solutions) if, and only if, d <0.
Examples:
-The solutions of the equation x² + x-2 = 0 are given by:
-The equation x²-4x + 4 = 0 has a repeated solution which is given by:
-The solutions of the equation x² + 1 = 0 are given by:
As can be seen in this last example, x2 is the conjugate of x1.
References
- Fuentes, A. (2016). BASIC MATH. An Introduction to Calculus. Lulu.com.
- Garo, M. (2014). Mathematics: quadratic equations.: How solve a quadratic equation. Marilù Garo.
- Haeussler, EF, & Paul, RS (2003). Mathematics for management and economics. Pearson Education.
- Jiménez, J., Rofríguez, M., & Estrada, R. (2005). Math 1 SEP. Threshold.
- Preciado, CT (2005). Mathematics Course 3rd. Editorial Progreso.
- Rock, NM (2006). Algebra I Is Easy! So Easy. Team Rock Press.
- Sullivan, J. (2006). Algebra and Trigonometry. Pearson Education.