The length of a rectangle is 22 cm and the width is 4 cm a similar rectangle has a width of 6 cm what is the perimeter of the second rectangle

The original rectangle measures 7 cm by 4 cm, and therefore has an area of

7cm x 4cm = 28cm^2

The new rectangle is 42cm^2 larger, therefore the new rectangle has an area of

28cm^2 + 42 cm^2 = 70cm^2

Since both sides of the original rectangle are increased by the same amount, the new rectangle can be described by the equation

(7cm + Xcm)(4cm + Xcm) = 70cm^2

Use FOIL (First, Outside, Inside, Last) to multiply the binomials, adding the results of the four multiplications

7cm x 4cm = 28cm^2
7cm x Xcm = 7Xcm^2
Xcm x 4cm = 4Xcm^2
Xcm x Xcm = X^2cm^2

Combine the like terms of 7X and 4X

X^2cm^2 + 11Xcm^2 + 28cm^2 = 70cm^2

To solve for X, bring all the terms to one side of the equation

X^2cm^2 + 11Xcm^2 + 28cm^2 - 70cm^2 = 0

X^2cm^2 + 11Xcm^2 - 42cm^2 =0

Now to solve for X we need to factor the lefthand side of the equation. We do this by finding two numbers that multiply to produce -42 and add to produce 11. Since the the result of multiplication is negative, and the result of addition is positive, we know that one of the factors will be negative, and the larger factor has to be positive. 

-1 and 42? No
-2 and 21? No
-3 and 14? Yes. 

Therefore factoring the equation gives us

(Xcm - 3cm)(Xcm + 14cm) = 0

The above equation equals 0, so we know that either 

(Xcm - 3cm) = 0

or

(Xcm + 14cm) = 0

Solve both and you get

X = 3 or X = -14

Since adding -14 to either of the sides of the original rectangle would produce a negative length or width, this cannot be the answer, therefore X = 3. We can check this by going back and adding 3 to each of the original dimensions. 

7cm + 3cm = 10cm and 4cm + 3cm = 7cm

10cm x 7cm = 70cm^2

This is 42cm^2 larger than the original rectangle, so we know we are correct.

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