Solving Quadratic Equations

Lesson Essential Question:How do you solve quadratic equations?

What does the solution(s) mean or represent?

Solving Quadratic Equations

What does it mean to solve a quadratic equation????

The "solutions" to the Quadratic Equation are where it is equal to zero.

Where: ax2 + bx + c = 0Where the graph crosses or touches the x-axis.They are also called "roots", or sometimes

"zeros"

Let’s take a look at where the solutions are and how many there can be!

Basketball Shot Revisited We used the function: h(t) = 8 + 40.8t – 16t2

to represent the height of the basketball as a function of the time the basketball was in the air after it was released from the players’ hands 8 feet above the ground.

Basketball Shot Revisited To find the time when the shot would reach

the 10 foot height of the basket, you would solve the equation:

8 + 40.8t – 16t2 = 10

To find the time when an “air ball” would hit the floor, you would solve the equation:

8 + 40.8t – 16t2 = 0

Basketball Shot Revisited 8 + 40.8t – 16t2 = 10 and 8 + 40.8t – 16t2 =

0The values of t that satisfy the equations are called the solutions of the equations.

For each situation, you could get good estimates of the solutions by searching in tables of values or by analyzing the graph of the function.

Let’s do that now. Let’s take a look at the graph and table of the function.

Let’s Start Getting Exact!!!

LEQ: What are some effective methods for solving quadratic equations algebraically to get exact solutions?

Do You Know???1. x2 = 36 What is the solution?2. x2 = 121 What is the solution?3. x2 = 100 What is the solution?4. x2 = 4 What is the solution?5. x2 = 64 What is the solution?6. x2 = 20 What is the solution?7. x2 = 55 What is the solution?8. X2 = -10 What is the solution?

Some quadratic equations can be solved by use of the fact that for any positive number n, the equation x2 = n is satisfied by two numbers:

n

n

Solving Quadratics • Solving a quadratic equation is similar to solving an equation• You want to get x all by itself, so you need to UNDO the

equation.• There may only be a squared term and a constant. (an ax2 and c

term)• Add or subtract the constant to its like term on the other side of

equals if needed.• Divide the a term on both sides of the equals.• Take the square root of both sides of the equals.• Declare your solution as both a positive and negative value.

Round to the nearest tenth if your solutions are a decimal. You cannot take the square root of a negative #, write no solution for this kind of problem.

4x2 = 100

2x2 + 6 = 18

5x2 + 40 = 20

-5x2 + 40 = 20

Practice the ConceptSolve the quadratic equations contained on the

worksheet, “Solving Simple Quadratic Equations”.

In each case, you will check your reasoning by substituting the proposed solution values for x into the original equation.

Platform High Diver

If the diver jumps off a 50-foot platform, what rule gives her or his distance fallen d (in feet) as a function of time t (in seconds)?

Think really, really, really hard…you’ve seen this equation before!!!!

Platform High Diver

The rule: d = 16t2

Write and solve an equation to find the time required for the diver to fall 20 feet.

Equation: 16t2 = 20

Determine the solution(s) to the quadratic equation.

16t2 = 20

Equation: 16t2 = 20

Can I have a volunteer to show their work and solution on the board?

Platform High Diver

What function gives the height, h, of the diver at any time, t, after she or he jumps from the 50-foot platform ?

Function: h(t) = 50 - 16t2

Platform High Diver

Write and solve an equation to find the time when the diver hits the water.

Equation: 50 - 16t2 = 0

Determine the solution(s) to the quadratic equation.

50 - 16t2 = 0

Equation: 50 - 16t2 = 0

Can I have a volunteer to show their work and solution on the board?

Factoring QuadraticsNow let’s move onto solving

quadratic functions by factoring.

Soccer PlayerIf a soccer player kicks the ball

from a spot on the ground with initial upward velocity of 24 ft/s, the height of the ball h (in feet) at any time, t, seconds after the kick will be approximated by the quadratic function:

h(t) = –16t2 + 24t

Soccer PlayerWhat time will the ball hit the

ground?Think about what your height is

on the ground?We must set the h(t) equal to zero

because that is the height from the ground.

0 = –16t2 + 24t

Soccer Player0 = 24t – 16t2

Be Careful!!!!This equation has to be solved differently than the square root method because it has the “bx” term in it!!

We must use our GCF skills!!

Soccer Player0 = 24t – 16t2

Let’s factor 24t – 16t2.

Factoring with GCF: 0 = 8t(3 – 2t)

Now we have to solve for t!!

Soccer Player0 = 24t – 16t2

The expression 8t(3 – 2t) will equal 0 when 8t = 0 and when 3 - 2t = 0.

Why?

Soccer Playerh(t) = 24t – 16t2

To find the solutions of the equation 24t – 16t2 = 0 we need to solve the factors!!

Solve 8t = 0 Solve 3 - 2t = 0.

Let’s Practice!Solve the following quadratic equations:1. 3x2 – 6x = 0

2. 4x2 + 2x = 0

3x2 – 6x = 03x(x – 2) = 0

3x = 0 and x – 2 = 0x = 0 and x = 2

4x2 + 2x = 02x(2x +1) = 0

2x = 0 and 2x + 1 = 0x = 0 and x = - ½

Let’s verify our answers by looking at the graph of the function.

Let’s verify our answers by looking at the graph of the function.

Let’s Practice Again!Solve the following quadratic equations:1. 7x2 + 14x = 0

2. 9x2 – 36 = 0

7x2 + 14x = 07x(x + 2) = 0

7x = 0 and x + 2 = 0x = 0 and x = -2

9x2 – 36 = 09(x2 – 4) = 0

9 = 0 and x2 – 4 = 0x = 2 and x = -2

Let’s verify our answers by looking at the graph of the function.

Let’s verify our answers by looking at the graph of the function.

Let’s Practice Once More!Solve the following quadratic equations:1. -x2 – 10x = 0

2. -4x2 – 12x = 0

-x2 – 10x = 0-x(x + 10) = 0

-x = 0 and x + 10 = 0x = 0 and x = -10

-4x2 – 12x = 0-4x(x + 3) = 0

-4x = 0 and x + 3 = 0x = 0 and x = -3

Let’s verify our answers by looking at the graph of the function.

Let’s verify our answers by looking at the graph of the function.

Let’s Practice Once More!Solve the following quadratic equations:1. x2 – 25 = 0

2. 16x2 – 576 = 0

x2 – 25 = 0 x2 = 25

x = 5 and x = -5

16x2 – 576 = 0 16x2 = 576

x2 = 36x = 6 and x = -6

Let’s verify our answers by looking at the graph of the function.

Let’s verify our answers by looking at the graph of the function.

Practice the ConceptSolve the quadratic equations contained on the

worksheet, “Solving Quadratic Equations by factoring out GCF”.

Check For UnderstandingComplete the Check For Understanding worksheet.

Ticket Out the DoorComplete the Ticket Out the Door assignment and

turn it in on your way out the door please.

Homework AssignmentComplete the “Solving Quadratic Equations Experts

Now” worksheet for homework please.