What if the discriminant of a quadratic equation is zero?

Q: What if the discriminant of a quadratic equation is 0?

A discriminant of zero indicates that the quadratic has a repeated real number solution. A negative discriminant indicates that neither of the solutions are real numbers.

Q: What happens if the discriminant is 0?

When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, there is exactly one real root.

Q: When the discriminant is zero What is the equation?

If the discriminant is equal to zero, this means that the quadratic equation has two real, identical roots. Therefore, there are two real, identical roots to the quadratic equation x2 + 2x + 1. D > 0 means two real, distinct roots.

The discriminant is the term underneath the square root in the quadratic formula and tells us the number of solutions to a quadratic equation. If the discriminant is positive, we know that we have 2 solutions. If it is negative, there are no solutions and if the discriminant is equal to zero, we have one solution. The discriminant is calculuated by squaring the "b" term and subtracting 4 times the "a" term times the "c" term.

Índice Show

What if the discriminant of a quadratic equation is 0?
What happens if the discriminant is 0?
When the discriminant is zero What is the equation?
When the discriminant is 0 How many solutions are there?

The discriminant is a really handy tool when you think you're getting a weird answer. Here's why. The discriminant tells you how many solutions there are to quadratic equation or how many x intercepts there are for a parabola. It doesn't tell you what those numbers are like what the x intercept values are, it just tells you how many of them there should be. And it sounds like that's not useful but it actually is especially when you're checking your work.
So here is what it looks like. The discriminant is the formula b squared minus 4ac remembering that a, b and c are the coefficients of your quadratic in standard form. It tells you the number of solutions to a quadratic equation. If the discriminant is greater than zero, there are two solutions. If the discriminant is less than zero, there are no solutions and if the discriminant is equal to zero, there is one solution.
This is something you just kind of have to memorize. It goes hand in hand with the quadratic formula. So if you guys have learned that, this will make a lot of sense. If you haven't learnt the quadratic formula yet you'll probably learn it tomorrow in math class. Just know that, what you're looking at is whether or not b squared minus 4ac is greater than zero, less than zero or equal to zero. And it tells me how many answers I should have. It doesn't tell me what the answers are just how many of them I should have in order to get the problem correct.

What is a discriminant?

A discriminant is a value calculated from a quadratic equation. It use it to 'discriminate' between the roots (or solutions) of a quadratic equation.

A quadratic equation is one of the form: ax2 + bx + c

The discriminant, D = b2 - 4ac

Note: This is the expression inside the square root of the quadratic formula

There are three cases for the discriminant;

Case 1:

b2 - 4ac > 0

If the discriminant is greater than zero, this means that the quadratic equation has two real, distinct (different) roots.

Example

x2 - 5x + 2 = 0

a = 1, b = -5, c = 2

Discriminant, D = b2 - 4ac

= (-5)2 - 4 * (1) * (2)

= 17

Therefore, there are two real, distinct roots to the quadratic equation

x2 - 5x + 2.

Case 2:

b2 - 4ac < 0

If the discriminant is greater than zero, this means that the quadratic equation has no real roots.

Example

3x2 + 2x + 1 = 0

a = 3, b = 2, c = 1

Discriminant, D = b2 - 4ac

= (2)2 - 4 * (3) * (1)

= - 8

Therefore, there are no real roots to the quadratic equation 3x2 + 2x + 1.

Case 3:

b2 - 4ac = 0

If the discriminant is equal to zero, this means that the quadratic equation has two real, identical roots.

Example

x2 + 2x + 1 = 0

a = 1, b = 2, c = 1

Discriminant, D = b2 - 4ac

= (2)2 - 4 * (1) * (1)

= 0

Therefore, there are two real, identical roots to the quadratic equation x2 + 2x + 1.

Summary

Quadratic equation is ax2 + bx + c

Determinant D = b2 - 4ac

D > 0 means two real, distinct roots.

D = 0 means two real, identical roots/

D < 0 means no real roots.

Now try these, (take care with minus signs)

Questions

Q1. x2 - 7x + 2 = 0

Q2. - 3x2 + 2x - 1 = 0

Q3. 9x2 - 12x + 4 = 0

Q4. - x2 + x + 1 = 0

Answers

Q1. D = 41, means two real, distinct roots.

Q2. D = -16, means no real roots.

Q3. D = 0, means two real, identical roots.

Q4. D = 5, means two real, distinct roots.

What if the discriminant of a quadratic equation is 0?

A discriminant of zero indicates that the quadratic has a repeated real number solution. A negative discriminant indicates that neither of the solutions are real numbers.

What happens if the discriminant is 0?

When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, there is exactly one real root.

When the discriminant is zero What is the equation?

If the discriminant is equal to zero, this means that the quadratic equation has two real, identical roots. Therefore, there are two real, identical roots to the quadratic equation x² + 2x + 1. D > 0 means two real, distinct roots.