Question

Does the parabola with equation x2 – x + 4 = 0 have real or imaginary roots?
1.
Real, because the discriminant is positive
2.
Real, because the discriminant is negative
3.
Imaginary, because the discriminant is positive
4.
Imaginary, because the discriminant is negative

Answer 1

4. If the determinant delta of a quadratic equation is positive, the equation has two real roots. If the determinant is negative, it has two imaginary roots. In this case, delta=b^2-4ac=(-1)^2-4*1*4=-15. Therefore, this equation has two imaginary roots.

Answer 2

Answer:

The parabola with equation $x^2\:-\:x\:+\:4\:=\:0$ has two imaginary roots, because the discriminant is negative.

Step-by-step explanation:

The quadratic formula says that the solutions are

                                                    $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

for any quadratic equation like:

                                                   $ax^2+bx+c=0$

The discriminant is the part of the quadratic formula under the square root.

The discriminant can be positive, zero, or negative, and this determines how many solutions there are to the given quadratic equation.

• A positive discriminant indicates that the quadratic has two distinct real number solutions.
• 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 but there are two imaginary roots that are complex conjugates.

We have the parabola with equation $x^2\:-\:x\:+\:4\:=\:0$

$\mathrm{For\:}\quad a=1,\:b=-1,\:c=4:\quad x_{1,\:2}=\frac{-\left(-1\right)\pm \sqrt{\left(-1\right)^2-4\cdot \:1\cdot \:4}}{2\cdot \:1}$

The discriminant for this equation is $\sqrt{\left(-1\right)^2-4\cdot \:\:1\cdot \:\:4}= \sqrt{-15}$, because the discriminant is negative the parabola has two imaginary roots.