Factoring Word Problems

A rectangle has a given are of: 8x2 + 2x - 15

a) Factor to find the algebraic expression for the length and the width of the rectangle.


A= 8x2 + 12x – 10x -15

A= 4x(2x+3) -5(2x+3)

A= (2x+3) (4x-5)

This is a complex trinomial meaning that the a and c were multiplied. Then two numbers were found that add up to the ac value and the numbers that multiply to it.


b) If x=12, determine the perimeter and area of the rectangle.


Area:

A= (2x+3) (4x – 5)

A= (2 x 12 +3) (4 x 12 – 5)

A= (24 + 3) (48 – 5)

A= (27) (43)

A= 1161 cm2


Perimeter:

P= 2l + 2w

P= 2(2x+3)+2(4x-5)

P= 4x + 6 + 8x - 10

P= 12x - 4

P= 12 x 12 -4

P= 144 - 4

P= 140 cm

Therefore, the area of the rectangle is 1161 cm and the perimeter is 140 cm.

The height of a ball thrown from the top of the building can be h= -5t2 - 15t -20), where t is the time in seconds, and h is the height in meters.

a) Write the formula in factored form.


When there is a question as such we find the GCF of the equation which in this case is -5. Once we have the GCF we take it out and divide it with the other terms. After dividing the terms we get new numbers in the brackets, in this case we have:

h=-5(t2 - 3t - 4)

The next step we do is look for numbers that add up to b and multiply to c. In our case -4+1 add up to -3, and -4x1 equal up to -4. After finding the numbers we factor by grouping. The answer that we end up with is:

h= -5(t-4) (t+1)


b) Use the factored form to find when the ball lands on the ground?

When the question is as such the h then changes to 0 because we want to know when the ball hits the ground meaning when it falls on the ground.

h= -5(t-4) (t+1)

0= -5(t-4) (t+1)

t-4=0 t+1=0

t= 4 t= -1

The signs change of the numbers because we are moving it to the other side of the equal sign.

The parabola has an equation of y= (x-4) (x+2)

a) Find the x intercepts

To find the x intercepts we set the y as 0 and solve for x.

y= (x-4) (x+2)

0= (x-4) (x+2)

x-4=0 x+2=0

x= 4 x=-2

Therefore the x-intercepts of the equation are (4,0) (-2,0). The r of the equation is 4, and the s is -2.


b) Find the vertex.

To find the vertex we have to find the Axis of symmetry (AOS) because it is the x of the vertex. To find the AOS we use the formula (r+s) /2. In our case it is

AOS= 4+(-2)/2

AOS= 2/2

AOS= 1

Vertex: (1,?)

To find the y we take the value of the AOS and substitute it in the formula. As shown:

y=(x-4) (x+2)

y= (1-4) (1+2)

y= (-3) (3)

y= -9

Vertex: (1, -9)

Therefore, the vertex of the equation y=(x-4) (x+2), is (1, -9).


If we were told to graph these three points then we would graph it by putting the values of r and s opposite from the point that we have gotten. In our case if we were to graph it we would graph the inteecepts first but we would change the signs. Instead of a positive 4 we graph it as -4 and the -2 as positive 2. Once we have our intercepts graphed we then will graph the vertex just as it is.