# Expressing the area of a living room in which the length is $10$ meters longer than its width

Redefining a problem by equating it:

A living room is $50m^2$. The length of the living room is $10$ meters longer than its width.

Surface: $length \times\ width$

Surface: = ($50m^2$)

$50 = (x + 10) \times\ x$

According to my book this can be simplified which makes sense. However, is the answer it provides a typo?

Final answer: $50 = x^2+10x$

Where does the $x^2$ come from? And isn't $20x$ to be read as $20 \times\ x$?

$$50 = (x+10)\cdot x$$

Knowing that multiplication is distributive, the answers should be:

$$50 = (x\cdot x) + (10\cdot x)\implies 50 = x^2+10\cdot x$$

The Distributivity Law states that:

$$(a±b)\cdot c = a\cdot c±b\cdot c$$

$$\frac{a±b}{c} =\frac{a}{c}+\frac{b}{c}$$

You use the distributive law to write $(x+10) \times x=x^2+10x$ The $x^2$ comes from the first factor. The $20$ should be $10$. Yes, $20x$ should be read as $20 \times x$

• Thanks, edited the OP. – yokihadu Aug 19 '17 at 22:38

A living room is $50m^2$.

This is the surface area.

The length of the living room is $10$ meters longer than its width.

This information relates two unknowns, the length and the width. So only one quantity of the two is really unknown and the other can be reconstructed using this relationship.

$50 = (x + 10) \times\ x$

We see that the sample solution chose to make the width the unknown, calling it $x$, and expressing the length as $x+10$.

Using the relationship that a rectangle has the area length times width, we indeed come up with the above equation.

According to my book this can be simplified which makes sense.

Yes, you can multiply it out.

$$50 = (x + 10) \times x = x \times x + 10 \times x = x^2 + 10 x$$

This is a quadratic equation in the unknown $x$, which can be rewritten as $$x^2 + 10 x - 50 = 0$$ The polynomial on the left hand side is of degree $2$.