Question

The area of a rectangular garden is 30 m?.

The length of the garden is (2x + 3) m and the width is (2 - 1) m.
a) Use the information above to write an equation of the form
ax? + ba + C= 0, where a, b and c are integers.
b) Work out the value of a to 1 d.p.

Answer 1

The equation is:

`2x^(2)+x-33=0`

While, the value of
`x` is:

`x=3.8`

Explanation:

Step 1: Creating the equation

The area for a rectangle is given by:

`A=l* w`

The
`A` is given as
`30`, and
`l` is equal to
`(2x+3)`, and
`w` is equal to
`(x-1)`.

Substitute these values into the formula:

`A=l* w\\30=(2x+3)* (x-1)`

Simplify the equation:

`30=(2x+3)* (x-1)\\\\\text{Use the distributive property for the expression on the right-hand side:}\\30=2x(x-1)+3(x-1)\\\\\text{Multiply:}\\30=2x^(2)-2x+3x-3\\\\\text{Subtract 30 from both sides of the equation:}\\30-30=2x^(2)-2x+3x-3-30\\\\\text{Simplify:}\\0=2x^(2)+x-33\\\\\text{Rearrange the equation to make it better looking:}\\2x^(2)+x-33=0`

Step 2: Find the value of x

We have the equation as:

`2x^(2)+x-33=0`

To find the value of
`x`, we can make use of the quadratic formula:

`x=\frac{-b \pm \sqrt{b^(2)-4ac}}{2a}`

From the equation, we have:

`a=2, b=1, c=-33`

Substitute these values into the formula:

`x=\frac{-1 \pm \sqrt{1^(2)-4(2)(-33)}}{2(2)}`

Calculate the values:

`x_(1)\approx 3.8, x_(2)\approx-4.3`

Since using a negative value of
`x`, can give negative lengths or widths (which is impossible), we will omit it.

Thus, the value of
`x` is:

`x=3.8`