please solve this problem with steps please

Question

 a parallelogram with a base of 2x+ 1 and a height of x + 3 has an area of 42 square cm. Find the base and height of the parallelogram. (HINT: A=bh)

Answer 1

Multiply the base by the height to get you the area.  Use the info given to write an equation.

 

(2x + 1)(x + 3) = 42

 

 

Expand out the left side.

 

2x² + 7x + 3 = 42

 

 

Subtract 42 on both sides of equation.

 

2x² + 7x - 39 = 0

 

 

Use the quadratic formula to solve for x.

 

x = (-b ± √(b² - 4ac)) / 2a

 

where:

a = 2

b = 7

c = -39

 

 

Plug in these values into the formula.

 

x = (-7 ± √(49 - 4(-78))) / 4

x = (-7 ± √(361)) / 4

x = (-7 ± 19) / 4

 

x = (-7 + 19) / 4            and            x = (-7 - 19) / 4

 

x = 3                            and             x = -6.5

 

 

Next, we validate these values by plugging them into the length and height.  If we plug in the negative value of x, both dimensions will be negative.  We cannot have a negative value for dimensions.

 

We accept x=3

 

length = 2(3) + 1 = 7

height = 3 + 3 = 6

 

7 × 6 = 42 , which is the area.

 

The length is 7 cm.

The height is 6 cm.

 

Answer 2

Peter, if A = bh and we have b and h, we need to multiply them!

 

42 = (2x + 1)(x + 3)

42 = 2x² + x + 6x + 3

42 = 2x² + 7x + 3

0 = 2x² + 7x - 39

 

Now we need to factor 2x² + 7x - 39.  We do this by multiplying 2 x -39, then finding two factors of the product that add to make 7.

 

2 x -39 = -78.  What are two factors of -78 that add together to make -78?  2 and -39? Nope.  3 and -16? Nope.  6 and -13?  They add to make -7, so we can switch the signs.  -6 + 13 = 7.

 

We will now split 7x into -6x + 13x and plug that into the equation.

 

0 = 2x² - 6x + 13x - 39

 

Next, we separate the first two terms from the 2nd two terms.

 

What is common among 2x² - 6x?  2x, right?  Factoring out, we get 2x(x - 3)

 

What is common amont 13x - 39? 13.  We get 13(x - 3)

 

Putting everything back in, we get 0 = 2x(x - 3) + 13(x - 3)

 

Just like 2x + 3x = 5x, 2(x + 1) + 3(x + 1) = 5(x + 1).  Since 2x and 13 both multiply with (x-3), we can add them together, although 2x and 13 do not combine, sooooo, we get...

 

0 = (2x + 13)(x - 3)

 

Using the zero factor rule, either 2x + 13 and/or x - 3 are equal to 0 for their product to be 0.

 

2x + 13 = 0

        2x = =13

          x = -6.5

 

We can't have a negative in this case, so that answer is out!

 

x - 3 = 0

     x = 3

 

Let's take our 3 and plug them in to find the base and height.

 

b = 2(3) + 1 = 7

h = 3 + 3 = 6

 

To check, 7 x 6 = 42, so it seems we have the right answer!