WritinginMathUsetheinformationon page 296toexplainhowadiversheightabovethepoolisrelatedtotime.Explainhowyoucoulddeterminehowlongitwilltakethedivertohitthewaterafterjumpingfromtheplatform.

5-6

The Quadratic Formula

and the Discriminant

Main Ideas

. Solve quadratic

equations by using

the Quadratic

Formula

. Use the discriminant

to delermine the

number and type of

rools of a quadratic

equation

GET READY for the Lesson

Competitors in the 10-meter platform

diving competition jump upward and

outward before diving into the pool

below. The height h of a diver in

meters above the pool after t seconds

can be approximated by the equation

h=-4.912 +31 + 10.

New Vocabulary

Quadratic Formula

discriminant

Quadratic Formula You have seen that exact solutions to some quadratic

equations can be found by graphing, by factoring, or by using the Square

Root Property. While completing the square can be used to solve any

quadratic equation, the process can be tedious if the equation contains

fractions or decimals. Fortunately, a formula exists that can be used to

solve any quadratic equation of the form ex2 + bx + c = 0. This formula

can be derived by solving the general form of a quadratic equation.

ax2 + bx+c=0

General quadratic equation

**+ **+ = 0

Divide each side by .

Subtract from each side

*+*+—+ Complete the square

62 – Lac

(x+ – bordo Factor the left side. Simplify the right side.

62 – Mac

*+

Square Root Property

62 – Mac

Subtract from each side

– Vb2 – 4oc

Simply

62

= f

20

YE

+

2x

24

I=

20

Reading Math

Quadratic Formula The

Quadratic Formula

is read Xeuws the

opposite of b, plus or

minus the square rot

of b squared mis doc

al divided by 20

This equation is known as the Quadratic Formula.

KEY CONCEPT

Quadratic Formula

The solutions of a quadratic equation of the form ax + bx + C = 0, where

00, are given by the following formula.

-b + VD2 – doc

20

276 Chapter 5 Quadratic Functions and Inequalities

Dr.

Introduction: In the field of mathematics, the study of quadratic equations is significant as it helps in solving problems related to the real-world scenarios. The height of a diver during a platform diving competition can be modeled using a quadratic equation. This brings us to the topic of the Quadratic Formula and the Discriminant, which is essential in solving any quadratic equation.

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Description: In this lesson, we will learn about the Quadratic Formula and the Discriminant and how they can be used to solve quadratic equations. The Quadratic Formula helps in solving any quadratic equation of the form ax^2 + bx + c = 0, which cannot be solved using other methods like factoring or completing the square. The Discriminant is used to determine the number and types of roots of a quadratic equation. This lesson also relates to real-world scenarios where quadratic equations are used in modeling various situations, like the height of a diver above a pool. By using the Quadratic Formula and the information given in such problems, it is possible to determine how long it will take for a diver to hit the water after jumping from the platform. Overall, this lesson will provide a deeper understanding of Quadratic Functions and Inequalities, which is a crucial topic in mathematics.

Objectives:

– To understand the concept of the Quadratic Formula and how it is used to solve quadratic equations of the form ax^2 + bx + c = 0.

– To understand how the discriminant is used in determining the number and type of roots of a quadratic equation.

– To apply the Quadratic Formula and the discriminant in solving real-world problems, particularly in finding the time it takes for a diver to hit the water after jumping from a platform.

Learning Outcomes:

By the end of the lesson, learners should be able to:

– Derive the Quadratic Formula and solve quadratic equations of the form ax^2 + bx + c = 0 using this formula.

– Determine the number and type of roots of a quadratic equation using the discriminant.

– Apply the Quadratic Formula and the discriminant in solving problems involving real-world situations, such as finding the time it takes for a diver to hit the water after jumping from a platform.

– Understand the connection between mathematics and other fields, such as physics and sports, in this case, diving.

Solution 1:

To determine how long it will take a diver to hit the water after jumping from the platform, we need to use the equation h(t) = -4.9t^2 + 31t + 10, where h is the height of the diver above the pool in meters and t is the time in seconds. We know that when the diver hits the water, the height h will be equal to zero, so we can set the equation equal to zero and solve for t using the quadratic formula:

-4.9t^2 + 31t + 10 = 0

t = (-31 ± sqrt(31^2 – 4(-4.9)(10)))/(2(-4.9))

t = (-31 ± sqrt(1241.6))/(-9.8)

t ≈ 3.16 seconds or t ≈ 5.1 seconds

Therefore, it will take about 3.16 seconds or 5.1 seconds for the diver to hit the water after jumping from the platform.

Solution 2:

We can also use the discriminant to determine the number and type of roots of the quadratic equation -4.9t^2 + 31t + 10 = 0. The discriminant is given by b^2 – 4ac, where a = -4.9, b = 31, and c = 10.

b^2 – 4ac = 31^2 – 4(-4.9)(10) = 1241.6

Since the discriminant is positive, the quadratic equation has two real roots. Also, since the discriminant is not a perfect square, the roots are irrational.

Using the quadratic formula, we can solve for the roots:

t = (-31 ± sqrt(1241.6))/(-9.8)

t ≈ 3.16 seconds or t ≈ 5.1 seconds

Therefore, the diver will hit the water after jumping from the platform in about 3.16 seconds or 5.1 seconds.

Suggested Resources/Books:

– Algebra and Trigonometry by Michael Sullivan

– Precalculus by James Stewart

– The Calculus Lifesaver: All the Tools You Need to Excel at Calculus by Adrian Banner

Similar asked questions:

1. How can the Quadratic Formula be used to solve any quadratic equation?

2. What is the discriminant and how is it used in quadratic equations?

3. In what situations might factoring or using the Square Root Property be more preferable than using the Quadratic Formula?

4. How can the height of a diver above a pool be related to time using quadratic equations and formulas?

5. What is the significance of the vertex in a quadratic equation and how can it be found using different methods?

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