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There are several methods for solving quadratic equations. In this lesson, the Quadratic Formula will be presented, proven, and used. Additionally, a method for determining the number of solutions without actually solving the equation will be presented.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Try a few practice exercises as a warm-up!
a Simplify the numeric expression by using the properties of square roots to remove the perfect-square factor.
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b Consider the quadratic function Of the following expressions, which represents the same function written in standard form?
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c Identify the coefficient of the quadratic equation when written in standard form
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Challenge
Is There a Solution?
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 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."
Discussion
The Quadratic Formula
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!
Rule
Quadratic Formula
The Quadratic Formula can be used to solve a quadratic equation written in standard form
Proof
The Quadratic Formula can be derived by completing the square given the standard form of the quadratic equation This method will be used to isolate the variable. To complete the square, there are five steps to follow.
1
Factor Out the Coefficient of
expand_more It is easier to complete the square when the quadratic expression is written in the form Therefore, the coefficient should be factored out.
Since the equation is quadratic, the coefficient is not equal to Therefore, both sides of the equation can be divided by
2
Identify the Constant Needed to Complete the Square
expand_more The next step is to rewrite the equation by moving the existing constant to the right-hand side. To do so, will be subtracted from both sides of the equation.
The constant needed to complete the square can now be identified by focusing on the term, while ignoring the rest. One way to find this constant is by squaring half the coefficient of the term, which in this case is
Note that leaving the constant as a power makes the next steps easier to perform.
3
Complete the Square
expand_more 4
Factor the Perfect Square Trinomial
expand_more The perfect square trinomial can now be factored and rewritten as the square of a binomial.
The process of completing the square is now finished.
5
Simplify the Equation
expand_more Finally, the right-hand side of the equation can be simplified.
Now, there is only one term. To isolate it is necessary to take square roots on both sides of the equation. This results in both a positive and a negative term on the right-hand side.
Now, the equation can be further simplified to isolate
Finally, the Quadratic Formula has been obtained.
▼
Simplify right-hand side
CommutativePropAdd
Commutative Property of Addition
CommutativePropMult
Commutative Property of Multiplication
SubFrac
Subtract fractions
Example
Solving a Quadratic Equation Using the Quadratic Formula
Magdalena will sell lottery tickets as a fundraiser to support paralympic athletes. The total profit depends on the price of a ticket and can be modeled by using the following quadratic equation.
Magdalena wants to raise at least However, she has not yet set the price of each lottery ticket. Help Magdalena find the smallest amount that can be charged per ticket and still make a profit of at least Round the price to the nearest whole dollar (the dollar sign is not necessary).
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Hint
Since the profit should be at least let be equal to Then, rewrite the quadratic equation in standard form. The equation can be solved using the Quadratic Formula.
Solution
It is given that the profit for the fundraiser should be at least Consider the given quadratic equation, which models the profit, and substitute for Then, rewrite the obtained equation in standard form.
Now, all of the coefficients in the standard form can be determined.
Therefore, and The obtained equation will be solved using the Quadratic Formula.
The values of and will now be substituted into the formula. Find by evaluating the right-hand side of the formula.
Using the Quadratic Formula, it was obtained that the solutions for the equation are Finally, both solutions can be evaluated using a table.
Since Magdalena wants the tickets to be as cheap as possible while making a profit of at least the price each ticket should be
Example
Solving a Quadratic Equation Not in Standard Form
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 feet above the ground.
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Hint
What is the height of the water stream's peak? Write a quadratic equation and solve it using the Quadratic Formula.
Solution
Consider the given situation on a coordinate plane and assume that the firefighter is standing on the axis. 
Since the water stream's peak is feet above the fire's base, whose height is feet, its height is feet. This height will now be substituted into the equation of the given quadratic function to calculate the desired distance.
The obtained quadratic equation can be solved using the Quadratic Formula. To do so, the equation must first be rewritten in standard form.
Next, the coefficients and can be identified.
Finally, these values will be substituted into the Quadratic Formula to solve the equation for
It has been found that this equation has exactly one solution, Therefore, it can be said that the firefighter is standing at a horizontal distance of feet from the water stream's peak.
SubstituteValues
Substitute values
▼
Evaluate right-hand side
CalcPow
Calculate power
SubTerm
Subtract term
CalcRoot
Calculate root
AddSubTerms
Add and subtract terms
CalcQuot
Calculate quotient
Pop Quiz
Solving a Quadratic Equation Using the Quadratic Formula
Solve the quadratic equations by using the Quadratic Formula. If necessary, round the answer to decimal places.
Discussion
Discriminant of a Quadratic Equation
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.
Concept
Discriminant
In the Quadratic Formula, the expression which is under the radical symbol, is called the discriminant.
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 |
|---|---|
Moreover, the discriminant determines the number of intercepts of the graph of the related quadratic function.
Example
Determining Whether There is a Solution Without Solving
A farmer wants to build a fence around a vegetable garden. To make it simple, the farmer will build it in the shape of a rectangle. The farmer has enough wood to build a fence the length of feet, including the gate.
The area of the vegetable garden changes as the side lengths of the rectangle change. The farmer wants the garden's area to be at least square feet. Will he need to buy more wood to achieve this goal?
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Hint
Let 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?
Solution
Let and be the side lengths of the rectangle that represents the vegetable garden. The length of the fence is the perimeter of the rectangle, which is the sum of all four side lengths.
It is given that the perimeter of the rectangle — how much wood the farmer has — is feet. By substituting for the equation can be solved for one of the side lengths, such as
The side lengths can now be placed in the diagram. It can be arbitrarily assumed that the length of the horizontal side is Keep in mind that both and are measured in feet.
Next, the area of the rectangle will be calculated in terms of The area of a rectangle is the product of the rectangle's length and width.
The obtained formula for is represented by a quadratic function. It is given that the farmer's desired area should be at least square feet. Therefore, this number will be substituted for in the formula.
The above is a quadratic equation that is not written in standard form. Hence, the equation will be rewritten to determine the number of solutions. Determining the number of solutions will help find if a value for exists so that the area of the rectangle is square feet.
The equation is now in standard form. This means that the number of solutions can be determined using the discriminant. Next, the coefficients and need to be identified.
The variables and can then be substituted into the discriminant
Since the discriminant is less than there are no solutions to the equation. Therefore, the farmer will not be able to build a fence so that the area of the vegetable garden is square feet. This means that he will need to buy more wood. Good thing he did the math before starting to construct the fence.
▼
Solve for
WriteDiffFrac
Write as a difference of fractions
CalcQuot
Calculate quotient
IdPropMult
Identity Property of Multiplication
RearrangeEqn
Rearrange equation
▼
Rewrite
Pop Quiz
Determining the Number of Real Solutions of a Quadratic Equation
Without solving the quadratic equations, use the discriminant to determine the number of real solutions.
Closure
Determining the Number of Solutions of a Quadratic Equation Without Solving
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 feet above the ground.
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Hint
Substitute for and identify the discriminant of the resulting quadratic equation.
Solution
Here, represents the number of seconds that passed since Diego threw the stone. Since the branch is situated feet above the ground, this number can be substituted for into the formula.
Although this quadratic equation can be solved using the Quadratic Formula, it is sufficient to find whether the equation has any solutions. To do so, the discriminant can be calculated. To calculate the discriminant, the equation needs to be rewritten in standard form.
Now, identify the coefficients and in the obtained equation.
Finally, the values of the coefficients will be substituted into the discriminant.
Because the value of the discriminant is greater than there are two solutions of the equation. Therefore, Diego and Magdalena know that the stone will reach the desired branch at two points in time.
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