Quadratic Word Problems

Eugine SzetoDarrell WorkmanEric ChenSani Patel

Maria's jardin

Maria is making a rectangular garden with the following layout. The length is 6 inches more than the width and the area of the rectangle is 91 square inches. Find the dimensions of the rectangle.

x

x+6

Explanation for Maria's JardinSince the length is 6 more than the width we let thewidth = x and the length = x + 6

x(x + 6) = 91

x2 + 6x = 91x2 + 6x - 91 = 0(x - 7)(x + 13) = 0

making the length 7 and the width -13(which is not valid for a rectangle) so it would be positive 13 instead.

You plant an apple tree in your backyard every year. Each year you continue to get better at planting and taking care of your tree. x being years since you started planting apple trees, the equation y=2x2+1x+25 represents the amount of apples you get from the tree each year.

1)according to the equation how many apples do you get from the tree on year 6?2)if x=0 represents your first year planting an apple tree, than how many apples did you get your first year? how does your answer relate to the graph?

Explanation

1) 2(6)2+1(6)+25= 2(36)+6+25= 72+6+25= 103-answer

2) 2(0)2+1(0)+25= 0+0+25= 25 in relation to the graph 25 represents the

y-intercept.-answer

Joe enjoys throwing spaghetti on his days off. He stands in his backyard every Saturday afternoon. His throwing arm is quite good, his record spaghetti toss was 12 feet in the air, after 4 seconds. At 2 seconds the spaghetti is 8 feet high. At 5 seconds it is at 4 feet. What is the equation of his throw and how high is the spaghetti when he throws it?

Joe's weekend

Explanation

Since you're given three points, you would want to use them. We are given (4,12) (2,8) and (5,4), thus we would plug them into x and y in the quadratic equation (ax2+bx+c) accordingly

16a+4b+c = 12 (4,12)4a+2b+c = 8 (2,8)25a+5b+c = 4 (5,4)Remember that in the first term, x is squared.

You can solve for one of the variables by using linear combination.

(16a + 4b + c = 12) - (4a + 2b + c = 8) = 12a + 2b = 4 (25a + 5b + c = 4 ) - (4a + 2b + c = 8) = 21a + 3b = -4

Then proceed to solve for one variable there. 3 ( 12a + 2b = 4 ) = 36a + 6b = 12 2 ( 21a + 3b = -4 ) = 42 a + 6b = - 8 resulting in -6a = -20 (a = 20/6) then

rinse and repeat for b and c to get y = -3.33x^2 +22x - 22.66.

However, matrices can also be used. You can use Cramer's Rule or the Inverse Matrix formula.

How to use Cramer's Rule: http://www.purplemath.com/modules/cramers.htmHow to find determinants: http://www.analyzemath.com/Tutorial-System-Equations/determinants.htmlax + by + cz = j

dx + ey + fz = kgx + hy + iz = l

x= y= z=

j b ck e fl h i

det A

a j cd k fg l i

det A

a b jd e kg h l

det A

16a + 4b + c = 124a + 2b + c = 825a + 5b + c = 4

x= y= z=

12 4 18 2 14 5 1

det A

16 12 14 8 125 4 1

det A

16 4 124 2 825 5 4

det A

16 4 14 2 125 5 1

-1

1284

* =abc

Use a calculator to solve.

Reminder on how to use the calculator with another example problem.

A B

-1

*

a= -10/3b= 22c= -68/3

The equation would be:y=-10/3x2+22x-68/3His starting height would be -68/3. (He is below ground!) This also the y-intercept when x or time is zero.http://www.youtube.com/watch?

feature=player_embedded&v=LAsECgE1eUs

Flight of a football1.) A quarterback throws a football to a

receiver at a height of 6 feet and a velocity of 34 feet per second. How many seconds does the receiver have to catch the football before it hits the ground?

2.) What is the highest height the ball will reach?

1.) Using the formula of a launched projectile, h=-16t2 + v0t + h0 we can determine model the flight of the object. h is the height. t is time. v0 is the initial velocity at which the object is launched, and h0 is the initial height of the object.

Our equation would be -16t2+34t+6. Then, using the quadratic formula, we can find the x-intercepts.

(2.88838554 , 0)

Using a calculator, we have two answers. -0.163838554 and 2.288838554. In the context of the problem, it has to be the positive answer because time cannot be negative.

To find the maximum height, or y, put 17/16 into the equation of the parabola.

x=

-b2a

x=

-(34)2(-16)

x= 17/16

2.) The highest point is the vertex. To find the vertex, use the formula:

y = -16 (17/16)2 + 34 (17/16) + 6y = 385/16 or 24.0625

*Note* When confronted with a quadratic equation and you are not good at factoring, you can always remember to use the quadratic formula.

Here are some ways to remember the quadratic formula.

http://www.youtube.com/watch?feature=player_embedded&v=O8ezDEk3qCg

http://www.youtube.com/watch?feature=player_embedded&v=U7q5fgGyqxk