Question

I need help with this math problem

Answer 1

AB = 4.5 cm

Explanation:

the total area (A) of the 2 rectangles is calculated as

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

= x² - 4x + 3x² - 6x

= 4x² - 10x

Given A = 36 , then equating

4x² - 10x = 36 ( subtract 36 from both sides )

4x² - 10x - 36 = 0 ( divide through by 2 )

2x² - 5x - 18 = 0 ← as required

To factorise the equation

consider the factors of the product of the coefficient of the x² term and the constant term which sum to give the coefficient of the x- term.

product = 2 × - 18 = - 36 and sum = - 5

the factors are + 4 and - 9

use these factors to split the x- term

2x² + 4x - 9x - 18 = 0 ( factor the first/second and third/fourth terms )

2x(x + 2) - 9(x + 2) = 0 ← factor out (x + 2) from each term

(x + 2)(2x - 9) = 0

equate each factor to zero and solve for x

x + 2 = 0 ⇒ x = - 2

2x - 9 = 0 ⇒ 2x = 9 ⇒ x = 4.5

but x > 0 , then x = 4.5

Then

AB = x = 4.5 cm

Answer 2

AB = 4.5 cm

Explanation:

`\boxed{\textsf{Area of a rectange}=\sf width * length}`

Area of the smaller rectangle:

`\implies A=x(x-4)`

`\implies A=x^2-4x`

Area of the larger rectangle:

`\implies A=(2x+x)(x-2)`

`\implies A=3x(x-2)`

`\implies A=3x^2-6x`

The area of the compound shape is the sum of the areas of the two rectangles:

`\implies A=(x^2-4x)+(3x^2-6x)`

`\implies A=x^2+3x^2-4x-6x`

`\implies A=4x^2-10x`

If the area of the compound shape equals 36 cm² then:

`\implies 36=4x^2-10x`

`\implies 36-36=4x^2-10x-36`

`\implies 0=4x^2-10x-36`

`\implies 4x^2-10x-36=0`

`\implies (4x^2)/(2)-(10x)/(2)-(36)/(2)=(0)/(2)`

`\implies 2x^2-5x-18=0`

The length of AB is x cm.

To find the value of x, factor the quadratic.

To factor a quadratic in the form
`ax^2+bx+c` find two numbers that multiply to
`ac` and sum to
`b`.

`\implies ac=2 \cdot -18=-36`

`\implies b=-5`

Therefore, the two numbers are: -9 and 4.

Rewrite
`b` as the sum of these two numbers:

`\implies 2x^2-9x+4x-18=0`

Factor the first two terms and the last two terms separately:

`\implies x(2x-9)+2(2x-9)=0`

Factor out the common term (2x - 9):

`\implies (x+2)(2x-9)=0`

Apply the zero-product property:

`(x+2)=0 \implies x=-2`

`(2x-9)=0 \implies x=(9)/(2)=4.5`

As length is positive, x = 4.5 only.

Therefore, AB = 4.5 cm.