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| 10 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Try a few practice exercises as a warm-up!
Magdalena and Diego, both huge fans of statistics, went camping to bond under the stars and talk stats. However, they realize that bears are in the area. They need to hang their food basket from a branch 15 feet above the ground. Diego figures he can throw a stone with a rope attached to it over the branch. As Diego winds up, Magdalena sheepishly snickers, "No way that works."
Besides graphing, using square roots, factoring, and completing the square, there is another method for solving a quadratic equation. This method consists of using the Quadratic Formula. Check out how to derive the formula by completing the square!
The Quadratic Formula can be used to solve a quadratic equation written in standard form ax2+bx+c=0.
x=2a-b±b2−4ac
2⋅2a=a
Commutative Property of Multiplication
a2+2ab+b2=(a+b)2
Commutative Property of Addition
(ba)m=bmam
(ab)m=ambm
ba=b⋅4aa⋅4a
Commutative Property of Multiplication
a⋅a=a2
Subtract fractions
ba=ba
a⋅b=a⋅b
a2=a
LHS−2ab=RHS−2ab
Put minus sign in numerator
Add and subtract fractions
x=2a-b±b2−4ac
Since the profit should be at least $200, let p(x) be equal to 200. Then, rewrite the quadratic equation in standard form. The equation can be solved using the Quadratic Formula.
p(x)=200
LHS−200=RHS−200
Rearrange equation
Substitute values
Calculate power
a(-b)=-a⋅b
(-a)(-b)=a⋅b
Subtract term
Calculate root
x=-4-32±16 | |
---|---|
x=-4-32+16 | x=-4-32−16 |
x=-4-16 | x=-4-48 |
x=4 | x=12 |
Since Magdalena wants the tickets to be as cheap as possible while making a profit of at least $200, the price each ticket should be $4.
A fire nozzle attached to a hose is a device used by firefighters to extinguish fires. Consider a firefighter who is aiming water to extinguish a fire on the third floor of a building. The base of the fire is situated 22 feet above the ground.
What is the height of the water stream's peak? Write a quadratic equation and solve it using the Quadratic Formula.
Distribute -0.008x
LHS−24=RHS−24
Rearrange equation
Substitute values
Calculate power
a(-b)=-a⋅b
(-a)(-b)=a⋅b
Subtract term
Calculate root
Add and subtract terms
-b-a=ba
Calculate quotient
Solve the quadratic equations by using the Quadratic Formula. If necessary, round the answer to 2 decimal places.
In general, quadratic equations have two, one or no real solutions. Before solving a quadratic equation, the number of real solutions can be determined by using the discriminant.
In the Quadratic Formula, the expression b2−4ac, which is under the radical symbol, is called the discriminant.
x=2a-b±b2−4ac
A quadratic equation can have two, one, or no real solutions. Since the discriminant is under the radical symbol, its value determines the number of real solutions of a quadratic equation.
Value of the Discriminant | Number of Real Solutions |
---|---|
b2−4ac>0 | 2 |
b2−4ac=0 | 1 |
b2−4ac<0 | 0 |
Moreover, the discriminant determines the number of x-intercepts of the graph of the related quadratic function.
Let x denote the length of one side of the rectangle. Then, use the fact that the length of the fence represents the perimeter of the rectangle. All things considered, how can the area of the rectangle be calculated?
P=800
LHS−2x=RHS−2x
LHS/2=RHS/2
Write as a difference of fractions
ca⋅b=ca⋅b
Calculate quotient
Identity Property of Multiplication
Rearrange equation
Distribute x
LHS−50000=RHS−50000
Commutative Property of Addition
Rearrange equation
Substitute values
Calculate power
a(-b)=-a⋅b
-a(-b)=a⋅b
Subtract term
Without solving the quadratic equations, use the discriminant to determine the number of real solutions.
The challenge presented at the beginning of this lesson asked if the stone thrown by Diego will reach, over some point in time, a branch located 15 feet above the ground.
Substitute 15 for h(t) and identify the discriminant of the resulting quadratic equation.
Substitute values
Calculate power
a(-b)=-a⋅b
-a(-b)=a⋅b
Subtract term
Enrique's profit from sales is his revenue R less his costs C. We can write this as the following function. V(p) = R(p) - C(p) We can describe the profit Enrique makes from his sales by finding an expression for the revenue and costs with respect to p. Keep in mind that Enrique sells every serving of ice cream he buys.
Because we have expressions for Enrique's costs and revenue, we can substitute these for R(p) and C(p) in our profit function. V(p) = px - 4x To write the function with respect to p only, we will eliminate x-variable by substituting x= 310-12p into the function rule and simplifying.
We wrote the function for Enrique's profit from sales in Part A. Notice that this is a quadratic function with a negative leading coefficient.
V(p) = - 12p^2+358p-1240
Therefore, the function reaches its maximum value at the vertex. The x-coordinate of the vertex is given by the equation of the axis of symmetry. We will write its equation by averaging the function's zeros. First we need to find them!
The profit is zero if Enrique sells the ice cream for 4 pesos or 1556 pesos. Next, we can calculate the mean of these values, p_1+p_22, to write the equation for the axis of symmetry of the parabola.
Enrique should sell his ice cream for about 15 pesos per serving to maximize his profit.