If the discriminant is negative how many solutions are there

The quadratic formula

Índice Show

The quadratic formula
The quadratic formula
All Signs Point to the Discriminant
Learning Outcome
The Discriminant
A General Note: The Discriminant
What happens when the discriminant is negative?
Does a negative discriminant mean no solutions?
How many solutions are there if the discriminant is positive?

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

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

Learning Outcome

Define the discriminant and use it to classify solutions to quadratic equations

The Discriminant

The quadratic formula not only generates the solutions to a quadratic equation, but also tells us about the nature of the solutions. When we consider the discriminant, or the expression under the radical, [latex]{b}^{2}-4ac[/latex], it tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.

Let us explore how the discriminant affects the evaluation of [latex] \sqrt{{{b}^{2}}-4ac}[/latex] in the quadratic formula and how it helps to determine the solution set.

If [latex]b^{2}-4ac>0[/latex], then the number underneath the radical will be a positive value. You can always find the square root of a positive number, so evaluating the quadratic formula will result in two real solutions (one by adding the positive square root and one by subtracting it).
If [latex]b^{2}-4ac=0[/latex], then you will be taking the square root of [latex]0[/latex], which is [latex]0[/latex]. Since adding and subtracting [latex]0[/latex] both give the same result, the “[latex]\pm[/latex]” portion of the formula does not matter. There will be one real repeated solution.
If [latex]b^{2}-4ac<0[/latex], then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.

The table below summarizes the relationship between the value of the discriminant and the solutions of a quadratic equation.

Value of Discriminant	Results
[latex]{b}^{2}-4ac=0[/latex]	One repeated rational solution
[latex]{b}^{2}-4ac>0[/latex], perfect square	Two rational solutions
[latex]{b}^{2}-4ac>0[/latex], not a perfect square	Two irrational solutions
[latex]{b}^{2}-4ac<0[/latex]	Two complex solutions

A General Note: The Discriminant

For [latex]a{x}^{2}+bx+c=0[/latex], where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers, the discriminant is the expression under the radical in the quadratic formula: [latex]{b}^{2}-4ac[/latex]. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.

Example

Use the discriminant to find the nature of the solutions to the following quadratic equations:

[latex]{x}^{2}+4x+4=0[/latex]
[latex]8{x}^{2}+14x+3=0[/latex]
[latex]3{x}^{2}-5x - 2=0[/latex]
[latex]3{x}^{2}-10x+15=0[/latex]

Example

Use the discriminant to determine how many and what kind of solutions the quadratic equation [latex]x^{2}-4x+10=0[/latex] has.

In the last example, we will draw a correlation between the number and type of solutions to a quadratic equation and the graph of its corresponding function.

Example

Use the following graphs of quadratic functions to determine how many and what type of solutions the corresponding quadratic equation [latex]f(x)=0[/latex] will have. Determine whether the discriminant will be greater than, less than, or equal to zero for each.

We can summarize our results as follows:

Discriminant	Number and Type of Solutions	Graph of Quadratic Function
[latex]b^{2}-4ac<0[/latex]	two complex solutions	will not cross the x-axis
[latex]b^{2}-4ac=0[/latex]	one real repeated solution	will touch x-axis once
[latex]b^{2}-4ac>0[/latex]	two real solutions	will cross x-axis twice

In the following video, we show more examples of how to use the discriminant to describe the type of solutions of a quadratic equation.

Summary

The discriminant of the quadratic formula is the quantity under the radical, [latex] {{b}^{2}}-4ac[/latex]. It determines the number and the type of solutions that a quadratic equation has. If the discriminant is positive, there are [latex]2[/latex] real solutions. If it is [latex]0[/latex], there is [latex]1[/latex] real repeated solution. If the discriminant is negative, there are [latex]2[/latex] complex solutions (but no real solutions).

The discriminant can also tell us about the behavior of the graph of a quadratic function.

What happens when the discriminant is negative?

If the discriminant is negative, that means there is a negative number under the square root in the quadratic formula. You may have learned in the past that you "can't take the square root of a negative number." The truth is that you can take the square root of a negative number, but the answer is not real.

Does a negative discriminant mean no solutions?

If the discriminant is negative, then the quadratic equation has no real solution. The discriminant is the expression b² – 4ac under the radical in the quadratic formula.

How many solutions are there if the discriminant is positive?

A Positive Discriminant If the discriminant is positive, this means that you have a positive number under the square root in the quadratic formula. This means you will end up with 2 real solutions. You can always take the square root of a positive number.