# Why a quadratic equation Cannot have one real root and one complex root?

## Why a quadratic equation Cannot have one real root and one complex root?

A quadratic equation may take complex values for x but the coefficients are always real. This makes both b and c complex, which is not allowed as they have to be real. This is the reason why if quadratic equations have complex roots, they are in pairs and form complex conjugates.

Can a quadratic equation have only one real root?

Thus, a parabola has exactly one real root when the vertex of the parabola lies right on the x-axis. The simplest example of a quadratic function that has only one real root is, y = x2, where the real root is x = 0.

How many complex roots does a quadratic equation have?

There are two complex roots. There are no x-intercepts. When graphing, if the vertex of the quadratic function lies above the x-axis, and the parabola opens upward, there will be NO x-intercepts and no real roots to the equation. The equation will have complex conjugate roots.

### Can there be one complex root?

The Fundamental Theorem of Algebra states that every polynomial of degree one or greater has at least one root in the complex number system (keep in mind that a complex number can be real if the imaginary part of the complex root is zero).

The discriminant is the expression b2 – 4ac, which is defined for any quadratic equation ax2 + bx + c = 0. 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.

Why is there only 1 real root when the discriminant is equal to zero?

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. D < 0 means no real roots.

#### How do you know if roots are real or 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.

How many roots real or complex does the polynomial 7 5x 4 3x 2 have in all?

Answer: All four roots are complex.

How do you know if a root is complex or real?

Real numbers have no imaginary part, and pure imaginary numbers have no real part. For example, if x = 7 is one root of the polynomial, this root is considered both real and complex because it can be rewritten as x = 7 + 0i (the imaginary part is 0).

## Are there two real roots?

For the quadratic equation ax2 + bx + c = 0, the expression b2 – 4ac is called the discriminant. The value of the discriminant shows how many roots f(x) has: – If b2 – 4ac > 0 then the quadratic function has two distinct real roots. – If b2 – 4ac = 0 then the quadratic function has one repeated real root.

How do you tell if a quadratic equation has no solution?

The first way to tell if a quadratic has no real solution is to look at the discriminant. If the discriminant is negative, then the quadratic equation has no real solution. Remember that for the quadratic equation ax2 + bx + c = 0, the discriminant is the expression b2 – 4ac.

Why does a quadratic equation have 1 solution?

Any time you end up with zero inside the square root of the Quadratic Formula, you’ll only get one solution to the equation, in the sense of getting one number that solves the equation.

### What happens when discriminant 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 are real and distinct roots?

If an equation has real roots, then the solutions or roots of the equation belongs to the set of real numbers. If the equation has distinct roots, then we say that all the solutions or roots of the equations are not equal. When a quadratic equation has a discriminant greater than 0, then it has real and distinct roots.

How do I find the root of one root?

Now, we can find the other root by the formula for sum and product of the roots. If $\alpha$ and $\beta$ are the two roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ then the sum and product of the roots are given by the formula: $\alpha +\beta =\dfrac{-b}{a}$ and $\alpha \beta =\dfrac{c}{a}$.

#### Are roots and zeros the same?

They are technically same, generally we use the word root for functions and zero for equations.

Can zeros be imaginary?

Complex zeros are values of x when y equals zero, but they can’t be seen on the graph. Complex zeros consist of imaginary numbers. An imaginary number, i, is equal to the square root of negative one.