4-5 Solving Quadratic equations

SOLVING quadratics

There are four ways to solve quadratics 1) Use factoring to solve 2) Find the solutions on a graph 3) Quadratic formula 4) completing the square

Solving quadratics by Factoring

ZERO-PRODUCT PROPERTY

If ab =0 , then a=0 or b=0

Steps to solve a quadratic by factoring:1.Factor ax2+bx+c=02.Set each factor equal to zero (zero-product property)3.Solve each equation

Solve the quadratic by factoring

062 xx0)3)(2( xx

0)3( x0)2( x

2x 3x

xxx 2214305 2 012325 2 xx0)25)(6( xx

0)25( x0)6( x

6x5

2x

xx 62

Solve by Graphing

The Solutions (a.k.a. zero, roots) to any quadratic is where the graph crosses the x-axis

Steps: 1. Graph quadratic (y2 = 0)

2. Find where the graph intersects the x-axis

Solve by Graphing

1572 2 xx

Set the equation equal to zero. Type into y1 & y2 = 0

Graph zoom 6 for standard window

Find where the quadratic intersects the x-axis. 2nd trace, intersect

X=1.5 & -5 X=-1.186141X= 1.6861407

These are called irrational numbers. Use Quadratic formula!!

42 2 xx

Solving quadratics by Quadratic formula You can solve any quadratic by using

the quadratic formula.

a

acbbx

2

42

Remember that “a” , “b” & “c” are the coefficients(numbers) from the quadratic

To Help you remember this formula:SongRAP?Please stop

Solve the quadratic by Quadratic formula

42 2 xx 02129 2 xx042 2 xx4

1

2

c

b

a

)2(2

)4)(2(4)1(1 2 x

4

331x

The number under the

radical is called the

discriminant

2

12

9

c

b

a

)9(2

)2)(9(4)12(12 2 x

18

21612x

This can be simplified further . We’ll get to that in

a later unit

RAP RAP strange

practice

Pg 245 #11-22 solve by using any method. Remember no irrational decimals!

Writing equations from roots Write a quadratic in standard form given

roots:

5,3

10,2

3

Solve quadratics To solve a quadratic, first try to factor or graph to

find the solutions. If unable to factor, or graph only gives irrational

decimals, then use quadratic formula.

132 xx0132 xx

)1(2

)1)(1(4)3(3 2

2

53

5912 2 xx05129 2 xx

)9(2

)5)(9(4)12(12 2

18

3241218

1812

3

1

3

5

Practice

Pg 245 #11-22

Using a Quadratic Equation The follow function models the path of a

kicked soccer ball. The height is y, the distance is x, and the units are meters.

1. How far does the soccer ball travel?

2. How high does the soccer ball go?

3. Describe a reasonable domain and range for the function.

xxy 60.103.0 2

Video of slip n slide The equation for the slip n slide is h(t) = -.12x2 + 1.6x +5 where H is the

height in feet and x is the distance from the end of the ramp. 1. How far does person

travel?

2. How high does the person go?

3. Describe a reasonable domain and range for the function.

YOU - TRY

A catapults that launches an object can be modeled with the equation y = -0.014x2 +0.68x + 10 where y is the height and x is the distance from the catapult. Units are in yards.

a.What is the max height of the object?b.How far did the object travel?c.How far was the object when it reached a height of 18 yards.

d.Video catapult (old Brit) video catapult(punkin chunkin)

Using a Quadratic EquationA volcanic eruption blasts a boulder upwards with an intitial velocity of 240 feet per second. How long will it take the boulder to hit the ground if it lands at the same elevation it was ejected? The path of the volcano can be modeled by h(t) = 240t – 16t2

Using a Quadratic Equation “The Celebrated Jumping Frog of Calaveras

County” written by Mark Twain in 1865, sparked the growth of frog-jumping competitions. The function models the height of one frog’s jump, where x is the distance, in feet, from the jump’s start.

xxy 59.029. 2

1. How far did the frog jump?

2. How high did the frog jump?

3. What is a reasonable domain and range for the frog-jumping function?

Video – NBA free-throw miss Video – JR HIGH

The height of a basketball is given by this function: h(x) = -0.125x2 + 1.75x + 7 where x is the time distance from the shooter and h is the height.

What was the max height of the shot?

If the player was 12 ft from the basket and the basket is 10 ft high)did the ball go in? ~did the ball pass the point (12, 10)

If a defender was 2ft from the shooter how high would they have to jump(straight up) in order to block the shot? ~when x is 2 how high was the ball?

Engineers can use the formula d = 0.05s2 + 1.1s to estimate the minimum stopping distance d in feet for a vehicle traveling s miles per hour.

a.If a car is traveling 45 mph what is the min. stopping distance?

b.If a car can stop after 65 feet, what is the fastest it could have been traveling?c.Video cars stopping