What does it mean if an equation has repeated roots?

A multiple root is a root with multiplicity , also called a multiple point or repeated root. For example, in the equation. , 1 is multiple (double) root. If a polynomial has a multiple root, its derivative also shares that root.

How do you find repeated roots?

From the above characterization of roots using the discriminant, we have the following:

  1. If 1 − 4 c > 0 , 1-4c > 0 , 1−4c>0, then the polynomial has two distinct real roots.
  2. If 1 − 4 c = 0 , 1-4c = 0, 1−4c=0, then the polynomial has a repeated root.

What does it mean when roots are imaginary?

We can think of the first term (½) as a starting place for finding the two roots. Then we see that the roots are located 3/2 from the starting point in both directions. This leads us to roots of a quadratic equation that does not cross the x-axis. These roots are known as complex (imaginary) roots.

How do you know if roots are imaginary?

Imaginary roots appear in a quadratic equation when the discriminant of the quadratic equation — the part under the square root sign (b2 – 4ac) — is negative. If this value is negative, you can’t actually take the square root, and the answers are not real.

Can a polynomial have repeated roots?

Whenever a part of the graph of a polynomial is in the form of a parabola whose vertex touches the x axis we conclude that a root is repeated at that point.

What is a repeated root in a quadratic equation?

When the left side factors into two linear equations with the same solution, the quadratic equation is said to have a repeated solution. We also call this solution a root of multiplicity 2, or a double root.

Can zero be a repeated root?

Repeated roots are for the same number, not “one positive, one negative”. We can see this in your example of x2(x−2)2=0, where there is a repeated root of x=2. Following from this, we can just as easily tell that x=0 is another repeated root (a good way to convince yourself is to write x2 as (x−0)2).

Why are imaginary roots important?

They are of enormous use in applied maths and physics. Complex numbers (the sum of real and imaginary numbers) occur quite naturally in the study of quantum physics. They’re useful for modelling periodic motions (such as water or light waves) as well as alternating currents.

Which equation has imaginary roots?

The roots belong to the set of complex numbers, and will be called “complex roots” (or “imaginary roots”). These complex roots will be expressed in the form a ± bi. A quadratic equation is of the form ax2 + bx + c = 0 where a, b and c are real number values with a not equal to zero.

Do imaginary roots always come in pairs?

The Complex Conjugate Root Theorem states that complex roots always appear in conjugate pairs.