# Math Help - Factorising quadratic equations

Hi

I have a question in my maths textbook that I can't figure out. It goes like this:

A rectangular garden has a square flower bed of side length x in one of it's corners. The remainder of the garden consists of lawn and has dimensions as shown. If the total area of the lawn is 50m²:
a) form an equation in terms of x,
b) solve the equation
c) calculate the length and width of the whole garden

Please, I'll be really grateful if someone could take the time to show me the workings of this sum.

2. Surely you can see that $\displaystyle l = x + 7$ and $\displaystyle w = x + 2$, so $\displaystyle A = (x + 7)(x + 2)$.

Since you're told the area is $\displaystyle 50\,\textrm{m}^3$, that means

$\displaystyle (x + 7)(x + 2) = 50$.

I'm sure you can solve this equation for $\displaystyle x$.

3. Also if you want to look at it simply,

the length is 5 bigger than the width, no matter what x is.
Hence, the positive factors of 50 that differ by 5 are 5 and 10.

The way the question is worded, you probably need to go Prove It's route
to satisfy the requirements of the question.

Alternatively, draw vertical and horizontal lines from the inner corner of the "x-box"
to the other sides of the garden and add up all the areas.

$x^2+7x+2x+7(2)=50$

$x^2+9x+14-14=50-14$

$x^2+9x=36$

$x(x+9)=36$

Now, you only need to find the positive factors of 36 that differ by 9.

4. Another thing to note is that dimensions can't be negative.............

5. If 'the remainder of the garden consists of lawn', then,
a) (x+2)(x+7) - x^2 = 50
b) x^2 + 2x + 7x + 14 - x^2 = 50
14x = 50 - 14
x= 36 / 14= 2.571
c) dimensions:
length= 2.571 + 7 = 9.571
width = 2.571 +2 = 4.571

6. Originally Posted by yorkey
Hi

I have a question in my maths textbook that I can't figure out. It goes like this:

Please, I'll be really grateful if someone could take the time to show me the workings of this sum.
Yikes !!!!!!

The garden consists of a lawn and an x-box flower bed.

The area of the lawn is $50\;m^2,$ not the entire garden.

There are 2 ways to form an equation in x

1. Write the area of the lawn, by drawing a horizontal or vertical inner line

$7x+(7+x)2=50$

or

$(2+x)7+2x=50$

2. Write the area of the entire garden

$(7+x)(2+x)=x^2+50$

Then you are solving

$7x+14+2x=50,$ or $14+7x+2x+x^2=x^2+50$

and both equations are the same.

Hence

$9x=50-14=36\Rightarrow\ x=4$

Finish with part (c) by adding x to 7 and to 2.