The quadratic formula
The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.
The quadratic formula
1 Answer
José F.
Mar 12, 2018
If you have a polynomial of the type #ax^2+bx+c#, the discriminant is #b^2-4ac#
Explanation:
Having a negative discriminant means that #b^2-4ac<0#, and the polynominal doesn't have real solutions.
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Solution:
A quadratic equation is an algebraic expression of the second degree in x.
The standard form of a quadratic equation is ax2 + bx + c = 0, where a, b are the coefficients, x is the variable, and c is the constant term.
Now let us consider a quadratic equation of the form of ax2 + bx + c.
According to formula, of the quadratic equation will be given as:
x = [-b ± √(b2 - 4ac)] / 2a
So, the discriminant of the quadratic equation is given as:
D = b2 - 4ac.
Then, we can see that if D < 0,
then x becomes an imaginary value.
Therefore, if D < 0, then √(b2 - 4ac) is a negative value and the roots will be imaginary.
If the discriminant of an equation is negative, which of the following is true of the equation.
Summary:
If the discriminant of an equation is negative, then the roots will be imaginary.
If the discriminant is negative the equation has solutions?
As stated in the attached link, there are three possible discriminant conditions: Positive, Zero, or Negative. If the discriminant is negative, there are no real solutions but there are two imaginary solutions. So, yes there are solutions if the discriminant is negative. The solutions are imaginary, which is perfectly acceptable as solutions.
The Discriminant (Jump to: Lecture | Video )
The Quadratic Formula is a method for solving quadratic equations. |
The Discriminant tells you how many real roots your quadratic equation has. |
If the discriminant is:
Positive, you have 2 real roots.
Zero, you have 1 real root.
Negative, you have 0 real roots(no solution).
All Signs Point to the Discriminant
Have you ever owned one of those Magic 8 Balls? They look like comically oversized pool balls, but have a flat window built into them, so that you can see what's insidea 20-sided die floating in disgusting opaque blue goo. Supposedly, the billiard ball has prognostic powers; all you have to do is ask it a question, give it a shake, and slowly, mystically, like a petroleum-covered seal emerging from an oil spill, the die will rise to the little window and reveal the answer to your question.
The quadratic equation contains a Magic 8 Ball of sorts. The expression b2 - 4ac from beneath the radical sign is called the discriminant, and it can actually determine for you how many solutions a given quadratic equation has, if you don't feel like actually calculating them. Considering that an unfactorable quadratic equation requires a lot of work to solve (tons of arithmetic abounds in the quadratic formula, and a whole bunch of steps are required in the completing the square method), it's often useful to gaze into the mystic beyond to make sure the equation even has any real number solutions before you spend any time actually trying to find them.
Talk the Talk
The discriminant is the expression b2 - 4ac, which is defined for any quadratic equation ax2 + bx + c = 0. Based upon the sign of the expression, you can determine how many real number solutions the quadratic equation has.
Here's how the discriminant works. Given a quadratic equation ax2 + bx + c = 0, plug the coefficients into the expression b2 - 4ac to see what results:
- If you get a positive number, the quadratic will have two unique solutions.
- If you get 0, the quadratic will have exactly one solution, a double root.
- If you get a negative number, the quadratic will have no real solutions, just two imaginary ones. (In other words, solutions will contain the i you learned about in Wrestling with Radicals.)
The discriminant isn't magic. It just shows how important that radical is in the quadratic formula. If its radicand is 0, for example, then you'll get
a single solution. If, however, b2 - 4ac is negative, then you'll have a negative inside a square root sign in the quadratic formula, meaning only imaginary solutions.
Example 4: Without calculating them, determine how many real solutions the equation 3x2 - 2x = -1 has.
Solution: Set the quadratic equation equal to 0 by adding 1 to both sides.
- 3x2 - 2x + 1= 0
You've Got Problems
Problem 4: Without calculating them, determine how many real solutions the equation 25x2 - 40x + 16 = 0 has.
Set a = 3, b = -2, and c = 1, and evaluate the discriminant.
- b2 - 4ac
- =(-2)2 - 4(3)(1)
- = 4 - 12
- = -8
Because the discriminant is negative, the quadratic equation has no real number solutions, only two imaginary ones.
Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
You can purchase this book at Amazon.com and Barnes & Noble.
- Algebra: The Quadratic Formula