Quadratic Formula Solutions for Complex Numbers

In summary, the solutions for the quadratic equation with complex coefficients are given by the formula ##x_1 = \frac{-b + \sqrt{b^2-4ac}}{2a}## and ##x_2 = \frac{-b - \sqrt{b^2-4ac}}{2a}##, where ##\Delta## is the discriminant and ##a, b, c \in \mathbb{C}##. The discriminant determines the nature of the solutions, with a complex discriminant indicating complex solutions. The quadratic formula can be used for both real and complex coefficients, but the use of complex solutions may depend on the application of the equation.
  • #1
Jhenrique
685
4
The solutions ##(x_1, x_2)## for the quadratic equation ##(0=ax^2+bx+c)##:

##x_1 = \frac{-b + \sqrt{b^2-4ac}}{2a}##

##x_2 = \frac{-b - \sqrt{b^2-4ac}}{2a}##

Are true if ##x## and ##a##, ##b##, and ##c## ##\in## ##\mathbb{C}## ?
 
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  • #4
The formula is variable in that the formula for x depends on the numbers a, b, and c. The discriminant is b^2-4ac. The way the discriminant relates to zero tells if there are two real solutions for x, or just one real solution for x, or two complex solutions for x using imaginary numbers.
 
  • #5
symbolipoint said:
The formula is variable in that the formula for x depends on the numbers a, b, and c. The discriminant is b^2-4ac. The way the discriminant relates to zero tells if there are two real solutions for x, or just one real solution for x, or two complex solutions for x using imaginary numbers.

But the solution showed in my first post is valed for Δ=0 and Δ<0?
 
  • #6
You mean delta as the discriminant? It is valid either way. If discriminant is less than zero, then , as I already said, x is not a real number. If a nonreal number makes no sense in a particular example application, then the solution for x is not valid.
 
  • #7
I rechecked the last part of your first message on the topic. What I said is mostly for typical college algebra/ intermediate algebra student. The next person who responds should be a member with much more knowledge about complex numbers.
 
  • #8
symbolipoint said:
I rechecked the last part of your first message on the topic. What I said is mostly for typical college algebra/ intermediate algebra student. The next person who responds should be a member with much more knowledge about complex numbers.

The wiki article says in plain type:
A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.

Doesn't that answer your question?
 
  • #9
Jhenrique said:
But the solution showed in my first post is valed for Δ=0 and Δ<0?

These conditions are for Real coefficients. The 2nd has no meaning if the coefficients are complex since the complex are not an ordered field. If the coefficients are complex then you would need to know how to handle the square root and do complex arithmetic, but it should yield your 2 roots.
 
  • #10
jedishrfu said:
The wiki article says in plain type:
A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.

..which is a particular case of the http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

To answer the OP's question, just substitute the given solutions in the equation and show they are correct.
 
  • #11
But ##\Delta## ##\in## ##\mathbb{C}## isn't necessary to apply the formula:

asd.png


in ##\frac{-b\pm \sqrt{b^2-4ac}}{2a}## ?
 
  • #12
AlephZero said:
..which is a particular case of the http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

To answer the OP's question, just substitute the given solutions in the equation and show they are correct.

Thanks for the more generalized answer. The OP initial post asked if x,a,b,c could be elements of Complex numbers and so I posted the article which he did not completely agree with and so I posted the specific statement from the wiki article on the quadratic formula in response.
 
  • #13
Solution by discriminant is not necessarily limited to real numbers, in fact it finds what values x can have,
now, which means that x can be either real or complex, the matter is where you use it. For example if you use it for real functions, it's best if you cross out a discriminant less than 0 or even complex since what you originally want is a real solution. Outside the real number limited exercises, you can use it and get the result you want. It does have some use in complex numbers, depending on what result you want to satisfy.
 

What is the quadratic formula?

The quadratic formula is a mathematical equation used to solve quadratic equations, which are equations with a single variable and a squared term. It is written as x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

When is the quadratic formula used?

The quadratic formula is used when solving quadratic equations, which can arise in various areas of mathematics and science. It is particularly useful in solving problems involving motion, such as projectile motion, and in finding the roots of a polynomial function.

What do the variables in the quadratic formula represent?

The variable x represents the solutions or roots of the quadratic equation. The coefficients a, b, and c represent the numbers that are multiplied by x^2, x, and the constant term, respectively.

How do you use the quadratic formula to solve an equation?

To use the quadratic formula, you first need to identify the values of a, b, and c in the quadratic equation. Then, substitute these values into the formula and solve for x. In most cases, the equation will have two solutions, which can be found by using both the + and - signs in the formula.

What are the applications of the quadratic formula?

The quadratic formula has many applications in various fields, including physics, engineering, and economics. It can be used to solve problems involving projectile motion, to determine maximum or minimum values of a function, and to find the roots of a polynomial equation. It is also used in computer graphics to create smooth curves and in finance to calculate compound interest.

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