Chapter 7: Quadratic Equations - Mr. McDonald's …mrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/math_2201...Chapter 7: Quadratic Equations Section 7.2 ... the standard form

Chapter 7: Quadratic Equations Section 7.2

Chapter 7: Quadratic Equations

Section 7.1: Solving Quadratic Equations by Factoring

Terminology:

Quadratic Equation:

A polynomial equation of the second degree; the standard form of a basic equation is

𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

Where a, b, & c are real numbers and 𝑎 ≠ 0.

Review: Factoring Quadratic Equations (from Gr.10)

Factoring A Quadratic Expression

A quadratic equation can be solved in many cases by factoring. There are four major

ways to factor a trinomial of the form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. These were covered in math

1201 so we shall do a quick review.

1. Factoring Using Product and Sum:

Product and Sum can only be used in situations where “a=1.” In such cases you

must determine your factors by concluding what possible combination of two

numbers can multiply to “c” and add to “b”

EXAMPLES: Factor The following

a. 𝑥2 + 6𝑥 + 8 b. 𝑘2 − 7𝑘 − 30

c. 𝑗2 + 11𝑗 − 42

d. 𝑓2 − 9𝑓 + 20


2. Factoring Using Decomposition:

Decomposition can be used in situations where “a≠1” and a GCF cannot be

removed. In such cases you must determine your factors by following these steps:

STEP1: Conclude what possible combination of two numbers can multiply to

“a×c” and add to “b.”

STEP2: Decompose your middle term into those two numbers.

STEP3: Group the first set and second set of terms. Pull out the GCF

(Greatest Common Factor) of each group.

STEP4: Then factor out the common bracketed term.


a. 5𝑥2 − 7𝑥 − 6 b. 3𝑘2 − 13𝑘 − 10

c. 8𝑗2 + 18𝑗 − 5 d. 15𝑓2 − 7𝑓 − 2

3. Factoring Using GCF:

In some cases where “a≠1,” a GCF can be removed from the situation and allow it

to be factored using Product and Sum or via Decomposition (with slightly more

manageable numbers).

EXAMPLES: Solve The following

a. 5𝑥2 − 10𝑥 b. 3𝑘2 − 9𝑘 − 12

c. 20𝑥2 − 50𝑥 − 30

d. −10𝑥2 + 14𝑥 + 12


4. Difference of Squares:

Difference of squares can only be used in situations where “b=0.” In such cases

both the “a” and “c” values will be perfect squares and there is a subtraction

symbol between them. Your resulting factors will be the square root of each term

with a different sign between them.


a. 𝑥2 − 9 b. 9𝑘2 − 25

c. 144𝑗2 − 100 d. 12𝑓2 − 75


Section 7.1: Solving Quadratic Equations by Factoring

Terminology:

Roots:

The values of the variable that make an equation in standard form equal to zero. These

are also called the solutions of the equation, the x-intercepts or the zeros of a graph.

These terms can and will be used interchangeably.

NOTE: To determine the roots of an equation, we must make sure first that it equals zero.

We then factor and set each factor equal to zero. This is called the zero product

principal in which if the product of two real numbers is zero, then one or both of the

numbers must be zero.

Ex. Solve each equation by factoring.

(a) 𝑥2 + 6𝑥 + 8 = 0 (b) 𝑘2 + 8𝑘 + 7 = 0

(c) 𝑗2 + 3𝑗 − 54 = 0 (d) 𝑧2 − 9𝑧 − 70 = 0

(e) 5𝑥2 − 7𝑥 − 6 = 0 (f) 4𝑘2 − 21𝑘 + 20 = 0

(g) 6𝑗2 + 18𝑗 = 0 (h) 63𝑥2 − 56𝑥 = 0


(i) 𝑥2 − 9 = 0 (j) 144𝑦2 − 100 = 0

(k) 49ℎ2 − 64 = 0 (l) 45𝑤2 − 80 = 0

(m) 1

2𝑚2 + 3𝑚 + 4 = 0 (n)

1

5𝑝2 − 2𝑝 + 5 = 0

(o) 0.25𝑞2 + 0.5𝑞 − 2 = 0 (p) 0.1ℎ2 − 1.2ℎ − 4.5 = 0


Word Problems Involving Factoring

Ex. The entry to the main exhibit hall in an art gallery is a parabolic arch. The arch can be modelled by the function:

ℎ(𝑤) = −0.625𝑤2 + 5𝑤 where the height, h(w), and width, w, are measured in feet. Several sculptures are going to be delivered to the exhibit hall in crates. Each crate is a square-based rectangular prism that is 7.5 ft high, including the wheels. The crates must be handled as shown, to avoid damaging the fragile contents. Determine the distance between the two points on the arch that are 7.5 ft high. Ex. Sanela sells posters to stores. The profit function for her business is:

𝑃(𝑛) = −0.25𝑛2 + 6𝑛 − 27 where n is the number of posters sold per month, in hundreds, and P(n) is the profit, in thousands of dollars. (ie if profit were $15 000, we would let 𝑃(𝑛) = 15). (a) If Sanela wants to earn a profit of $5000 how many posters must she sell? (b) If Sanela wants to earn a profit of $9000, how many posters must she sell?


Determining a Possible Quadratic Equation Given its Roots Ex. Tori says she solved a quadratic equation by graphing it. She says that the roots were −5 and 7. Determine the quadratic equation that she may have solved. Ex. Write a quadratic function that has zeros at 0.5 and -0.75. Ex. The x-intercepts of a quadratic function are 3 and −2.5. Write a quadratic equation that has these roots.


Section 7.2: Solving Quadratic Equations by Graphing

Determining Solutions of a Quadratic Equation

Determining the solutions from a graph is simple as you only need to identify the locations of

the x-intercepts taken directly from the graph

Ex. Determine the roots of each quadratic function


Section 7.3: Using Quadratic Formula

Terminology:

Quadratic Formula:

A formula for determining the roots of a quadratic equation in the form

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎 ≠ 0. The quadratic formula is written using the

coefficients of the variables and the constant in the quadratic equation that is

being solved:

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

Inadmissible Solution:

A root of a quadratic equation that does not lead to a solution that satisfies the

original problem.

Solving Quadratic Equations Using the Quadratic Formula

(a) 6𝑥2 − 7𝑥 − 3 = 0 (b) 10𝑥2 − 3𝑥 − 18 = 0


(c) 12𝑥2 − 17𝑥 − 40 = 0

(d) −20𝑝2 + 7𝑝 + 3 = 0

(e) 1

2𝑧2 − 5𝑧 + 17 = 0


Word Problem Applications

Ex. A student council is holding a raffle to raise money for a charity fund drive. The

profit function for the raffle is:

𝑝(𝑐) = −25𝑐2 + 500𝑐 − 350

Where 𝑝(𝑐) is the profit and 𝑐 is the price of each ticket, both in dollars.

(a) What ticket price would result in the student council raising $700 to donate?

(b) What ticket price would result in the student council breaking even?


Ex. A rocket is launched into the air. Its motion can be modeled by the equation:

ℎ(𝑡) = −5𝑡2 + 37𝑡 + 20

where ℎ(𝑡) is the height in metres above the ground and 𝑡 is time in seconds.

(a) Determine when the rocket first reaches a height of 80 metres?

(b) How long does it take for the rocket to land back on the ground?


x

x

x

x

x

x

x

x

11 cm

8 cm

Ex. A photograph 8 cm by 11 cm will be framed as shown in the diagram. The combined area of the frame and photograph will be 180 cm2. Algebraically determine the outside dimensions of the frame.

Ex. A rectangular swimming pool has length 30 m and width 20 m. There is a deck of uniform width surrounding the pool. The area of the pool is the same as the area of the deck. Write a quadratic equation to model this situation and use it to determine the width of the deck.

x

x

x

x

x

x

x

x

30 m

20 m


Ex. A store rents an average of 750 video games each month at the current rate of $4.50. However,

for every $1 increase, they know that they will rent 30 fewer games each month. Create a quadratic

equation in standard form to represent the relationship and use it to determine the minimum number

of times the owners should increase the rental rate enough to generate revenue of $5000 per month?

Ex. The Math 2201 class is holding a bake sale to raise money for our class trip to PiWorld, the most exciting, most amazing, most interesting, most awesome Theme Park on the planet. They know that if they charge $7 per tray of cookies they will sell 400 trays. Their statistical analysis of the population has helped them to conclude that for every $1 they increase their price they will sell 30 less trays. Write a quadratic equation in standard form to represent this relationship. Use your equation to find out the minimum number of price increase they should make to reach their fundraising goal of $3200?

Documents

Chapter 7: Quadratic Equations - Mr. McDonald's …mrmcdonaldshomepage.weebly.com/uploads/1/3/9/2/13921997/math_2201...Chapter 7: Quadratic Equations Section 7.2 ... the standard form