c The discriminant of a quadratic equation is b^2-4ac.
D
d The discriminant of a quadratic equation is b^2-4ac.
A
a No real solutions
B
b Two real solutions
C
c One real solution
D
d Two real solutions
Practice makes perfect
a We want to determine the number of real solutions of the given quadratic equation.
(x-2)^2= - 3
Notice that there is no real number that results in a negative number when squared. Since - 3 <0, there is no real number that can satisfy the given quadratic equation. Therefore, this equation has no real solutions.
b We want to determine the number of real solutions of the given quadratic equation. To do this we will use the discriminant, which in the Quadratic Formula is b^2-4ac.
ax^2+bx+c=0
⇕
x=- b±sqrt(b^2-4ac)/2aIf we just want to know the number of real solutions and not the solutions themselves, we only need to work with the discriminant.
Since our equation is already given in standard form, let's first identify the values of a, b, and c.
6x^2-x-2=0
⇕
6x^2+( - 1)x+( - 2)=0
Finally, let's evaluate the discriminant.
Since the discriminant is 49, the quadratic equation has two real solutions.
Extra
Further information
If the discriminant is greater than zero, the equation will have two real solutions. If it is equal to zero, the equation will have one real solution. Finally, if the discriminant is less than zero, the equation will have no real solutions.
c We want to determine the number of real solutions of the given quadratic equation. To do this we will use the discriminant, which in the Quadratic Formula is b^2-4ac.
ax^2+bx+c=0
⇕
x=- b±sqrt(b^2-4ac)/2aIf we just want to know the number of real solutions and not the solutions themselves, we only need to work with the discriminant.
Since our equation is already given in the standard form, let's first identify the values of a, b, and c.
4x^2-4x+1=0
⇕
4x^2+( - 4)x+ 1=0
Finally, let's evaluate the discriminant.
Since the discriminant is 0, the quadratic equation has one real solution. At the end of Part B you can find further explanation.
d We want to determine the number of real solutions of the given quadratic equation. To do this we will use the discriminant, which in the Quadratic Formula is b^2-4ac.
ax^2+bx+c=0
⇕
x=- b±sqrt(b^2-4ac)/2aIf we just want to know the number of real solutions and not the solutions themselves, we only need to work with the discriminant.
Since our equation is already given in the standard form, let's first identify the values of a, b, and c.
427x^2+731x-280=0
⇕
427x^2+ 731x+( - 280)=0
Finally, let's evaluate the discriminant.
