The discriminant of a quadratic equation is b^2-4ac.
Quadratic Equation: - 5x^2+20x-50=0
No, there are no real number solutions of the equation.
Practice makes perfect
We will start by writing a quadratic equation modeling our earnings. Then, we will find the discriminant of the equation to check if it has any real solution.
Writing the Quadratic Equation
If we do not decrease the fee, we earn $700 in a week.
14 * 50 =700
We know that when we decrease the fee one dollar, we get 5 more costumers. Let x be the number of $1 decrease in our fee. Then, the expression below represents the amount of money we can earn per week.
(14- x)(50+ 5 x)
To see if we can ever earn $750, we need to to equate the expression and 750.
(14-x)(50+5x)= 750
Let's apply the Distributive Property and simplify it.
This quadratic equation, written in standard form, represents the situation.
Discriminant of the Equation
We want to use the discriminant of the quadratic equation to determine the number of real solutions.
In the Quadratic Formula, b^2-4ac is the discriminant.
ax^2+bx+c=0
⇕
x=- b±sqrt(b^2-4ac)/2a
If we just want to know the number of real solutions, and not the solutions themselves, we only need to work with the discriminant. Let's first rewrite the given equation in standard form.
Let's identify the values of a, b, and c.
- 5x^2+20x-50=0
⇕
- 5x^2+ 20x+( - 50)=0
Finally, let's evaluate the discriminant.
Since the discriminant is - 600, the quadratic equation has no real solution. That is, we cannot earn $750 in a week.
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.
