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### Course: Class 10 (Old)>Unit 4

Lesson 4: Nature of roots

# Discriminant for types of solutions for a quadratic

Discriminant for Types of Solutions for a Quadratic. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• At , what are conjugates?
• For any (a+b), its conjugate is (a-b) and vice versa. This is very handy when dealing with roots and complex numbers :)
• Under what circumstance is there only one complex solution?
• There are no circumstances in which you have one and only complex solution.
you either have two real solution or two complex solutions. This is because complex solution occur in pairs for example, a+bi or a-bi.
• The last video (Discriminant of Quadratic Equations) you wrote b² - 4ac < 0 is called NO REAL SOLUTION at , but in this video, why it's called 2 complex solutions? In the same previous video, you wrote b² - 4ac > 0 is called solutions, but you wrote b² - 4ac > 0 is called 2 real roots in this video. What is the difference?
• By definition complex numbers are "not real", so the two statements are equivalent. If an equation has real solutions it just means that its graph crosses the x axis at some point. If it doesn't have real solutions, the graph of the equation never crosses the x axis. This is because you're solving for x when y equals zero (notice quadratics are always solved equal to zero?), and since there IS NOT an x intercept in an equation that doesn't cross the x axis (sounds obvious I know) you won't get a solution.
There is more confusion about this than there has to be. "Imaginary solutions", "double roots", "conjugate pairs" and all that stuff sounds mysterious, but it only ends up that way because you're trying to find a relationship between a y value of zero, and x. The coordinate for a real number solution would be (x,0), so if there is no x intercept it's game over. But the graph of the equation still exists, you can see it right there on the paper. Sorry if this made no sense, just keep watching the vids, there are plenty on imaginary and complex numbers.
• How do you find the b if the equation is= 7y squared-2=0
• Andrew,
7y² - 2 = 0
And what is the coefficient of y?
I refer to it as being invisible. But you can make it visible by re-writing the equation as
7y² + 0y -2 = 0
So b = 0 when you use the quadratic formula for this equation.
• What would you do if you were trying to solve the discriminant of a quadratic equation with more than one variable and more than 3 terms? (For example x^2-2ax+2a^2+1=0)
• We just do exactly the same thing. You can lump the final two terms together, as neither of them involves x. So here, A = 1, B = -2a and C = 2a^2 + 1 (I've used capital letters for A, B and C since you've already used the variable 'a' in your quadratic). The discriminant is B^2 - 4AC, which is (-2a)^2 - 4(2a^2+1) = 4a^2 - 8a^2 - 4 = -4(a^2 + 1). What does this mean? Well, a^2 is always non-negative (i.e. zero or positive) so the bracketed term a^2+1 is always positive. Thus the discriminant is always -4 times a positive number, so it is always negative. Hence, whatever the value of 'a' in the equation you gave, the discriminant will be negative and the roots will be complex.
• Is there a difference between the roots and the solution?
• They are the same thing, but because distances cannot be negative, sometimes only one root works
(1 vote)
• How can you determine if the solution will have two irrational answers?
• How do I know whether there are two complex solutions or one complex solution??
• If the discriminant is negative, then there are always two complex solutions. In fact, if one is a + bi, then then other will always be a - bi.
• in previous video, it said if discriminant (b^2-4ac) < 0 : there is no solutions.
and it says here : two complex solutions ?!
(1 vote)
• There ARE solutions if the domain extends over all types of numbers. That is, if the domain includes complex numbers, not just real numbers.
• Can someone give examples of Complex Solutions (Including the conjugate)?
I'm imagining solutions of 2+3i and 2-3i or 1+√3i and 1-√3i.. Is this accurate?