Quadratic formulas and discriminants

Question

Look at the equation below.
X^2-6x+13=0
what is the value of the discriminant of the equation? WHAT does the value of the discriminant TELL YOU about the solutions to this equation? and determine the solutions to the equation.

Answer 1

ax² + bx + c = x² - 6x + 13

 

a = 1

b = -6

c = 13

 

D = b² - 4ac = (-6)² - 4(1)(13) = -16

 

D < 0 ⇒ There are two complex roots, and one is the complex conjugate of the other.

 

r = (-b ± √D)/(2a) = [-(-6) ± √(-16)] / 2 = (6 ± 4i) / 2

 

r₊ = 3 + 2i

r_- = 3 - 2i

 

Note the roots are complex conjugates of each other.

Answer 2

The discriminant is b^2 - 4ac (not so coincidentally, the square root of this is found in the quadratic formula); when the value is positive there are 2 roots; when it equals 0, there is one root of multiplicity 2; when it is negative, there are no real solutions (the solutions are complex).

In this case (-6)^2 - 4*1*13 = 36 - 52 = -16, so there are no real roots;

Using the quadratic formula, we see that the roots are (-(-6) +- sqrt((-6)^2 - 4*1*13 ))/2 = 3 +- 2 i

Answer 3

Given an equation in the form ax² + bx + c = 0, the discriminant is 

 

b² - 4ac

 

In this problem, a = 1, b = -6, c = 13

 

 

 

 

Answer 4

The discriminant d of the quadratic equation ax²+ bx +c=0 is

d = b²-4ac.

In this case a=1, b=-6 and c=13.

 

If d>=0 (greater or equal to zero) then the quadratic equation has solutions in the Real numbers axis.

If d<0 (less than zero) then it has complex solutions.

 

The quadratic equation has two roots or solutions:

 

x1=(-b+square root of discriminant)/2a

x2=(-b-square root of discriminant)/2a

 

In this case the solutions are complex.

Remember that square root of -1 = i and square root of (a*b) = (square root of a) * (square root of b)

 

Answer 5

x^2-6x+13=0

the discriminant is b^2-4ac

b^2-4ac=(-6)^2-4(1)(13)=36-52=-16

because the discriminant is negative there are two complex roots

if the discriminant is positive there are two real roots; two rational roots or two irrational roots depending on the discriminant being either a perfect square (rational) or not a perfect square (irrational)

if the discriminant is equal to 0 there are two equal double real roots

(-(-6)±√-16)/2=(6±4i)/2=3±2i are the roots